{"id":456,"date":"2021-02-04T15:27:52","date_gmt":"2021-02-04T15:27:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=456"},"modified":"2021-03-30T15:56:20","modified_gmt":"2021-03-30T15:56:20","slug":"problem-set-the-limit-of-a-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-the-limit-of-a-function\/","title":{"raw":"Problem Set: The Limit of a Function","rendered":"Problem Set: The Limit of a Function"},"content":{"raw":"
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For the following exercises (1-2), consider the function [latex]f(x)=\\dfrac{x^2-1}{|x-1|}[\/latex].<\/p>\r\n\r\n

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\r\n\r\n1. [T]<\/strong> Complete the following table for the function. Round your solutions to four decimal places.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/th>\r\n[latex]f(x)[\/latex]<\/th>\r\n<\/th>\r\n[latex]x[\/latex]<\/th>\r\n[latex]f(x)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
0.9<\/td>\r\na.<\/td>\r\n<\/td>\r\n1.1<\/td>\r\ne.<\/td>\r\n<\/tr>\r\n
0.99<\/td>\r\nb.<\/td>\r\n<\/td>\r\n1.01<\/td>\r\nf.<\/td>\r\n<\/tr>\r\n
0.999<\/td>\r\nc.<\/td>\r\n<\/td>\r\n1.001<\/td>\r\ng.<\/td>\r\n<\/tr>\r\n
0.9999<\/td>\r\nd.<\/td>\r\n<\/td>\r\n1.0001<\/td>\r\nh.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
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2.\u00a0<\/strong>What do your results in the preceding exercise indicate about the two-sided limit [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex]? Explain your response.<\/p>\r\n[reveal-answer q=\"fs-id1170571657469\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571657469\"]\r\n

[latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex] does not exist because [latex]\\underset{x\\to 1^-}{\\lim}f(x)=-2 \\ne \\underset{x\\to 1^+}{\\lim}f(x)=2[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises (3-5), consider the function [latex]f(x)=(1+x)^{1\/x}[\/latex].\r\n

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3. [T]<\/strong> Make a table showing the values of [latex]f[\/latex] for [latex]x=-0.01, \\, -0.001, \\, -0.0001, \\, -0.00001[\/latex] and for [latex]x=0.01, \\, 0.001, \\, 0.0001, \\, 0.00001[\/latex]. Round your solutions to five decimal places.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/th>\r\n[latex]f(x)[\/latex]<\/th>\r\n<\/th>\r\n[latex]x[\/latex]<\/th>\r\n[latex]f(x)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
\u22120.01<\/td>\r\na.<\/td>\r\n<\/td>\r\n0.01<\/td>\r\ne.<\/td>\r\n<\/tr>\r\n
\u22120.001<\/td>\r\nb.<\/td>\r\n<\/td>\r\n0.001<\/td>\r\nf.<\/td>\r\n<\/tr>\r\n
\u22120.0001<\/td>\r\nc.<\/td>\r\n<\/td>\r\n0.0001<\/td>\r\ng.<\/td>\r\n<\/tr>\r\n
\u22120.00001<\/td>\r\nd.<\/td>\r\n<\/td>\r\n0.00001<\/td>\r\nh.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
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4.\u00a0<\/strong>What does the table of values in the preceding exercise indicate about the function [latex]f(x)=(1+x)^{1\/x}[\/latex]?<\/p>\r\n[reveal-answer q=\"fs-id1170571654990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571654990\"]\r\n

[latex]\\underset{x\\to 0}{\\lim}(1+x)^{1\/x}=2.7183[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>To which mathematical constant does the limit in the preceding exercise appear to be getting closer?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (6-8), use the given values of [latex]x[\/latex] to set up a table to evaluate the limits. Round your solutions to eight decimal places.<\/p>\r\n\r\n

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6. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\dfrac{\\sin 2x}{x}; \\, x = \\pm 0.1, \\, \\pm 0.01, \\, \\pm 0.001, \\, \\pm 0.0001[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/th>\r\n[latex]\\frac{\\sin 2x}{x}[\/latex]<\/th>\r\n[latex]x[\/latex]<\/th>\r\n[latex]\\frac{\\sin 2x}{x}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
\u22120.1<\/td>\r\na.<\/td>\r\n0.1<\/td>\r\ne.<\/td>\r\n<\/tr>\r\n
\u22120.01<\/td>\r\nb.<\/td>\r\n0.01<\/td>\r\nf.<\/td>\r\n<\/tr>\r\n
\u22120.001<\/td>\r\nc.<\/td>\r\n0.001<\/td>\r\ng.<\/td>\r\n<\/tr>\r\n
\u22120.0001<\/td>\r\nd.<\/td>\r\n0.0001<\/td>\r\nh.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"fs-id1170571586213\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571586213\"]\r\n

a. 1.98669331; b. 1.99986667; c. 1.99999867; d. 1.99999999; e. 1.98669331; f. 1.99986667; g. 1.99999867; h. 1.99999999; [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin 2x}{x}=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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7. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\dfrac{\\sin 3x}{x}; \\, x = \\pm 0.1, \\, \\pm 0.01, \\, \\pm 0.001, \\, \\pm 0.0001[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
X<\/em><\/th>\r\n[latex]\\frac{\\sin 3x}{x}[\/latex]<\/th>\r\n[latex]x[\/latex]<\/th>\r\n[latex]\\frac{\\sin 3x}{x}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
\u22120.1<\/td>\r\na.<\/td>\r\n0.1<\/td>\r\ne.<\/td>\r\n<\/tr>\r\n
\u22120.01<\/td>\r\nb.<\/td>\r\n0.01<\/td>\r\nf.<\/td>\r\n<\/tr>\r\n
\u22120.001<\/td>\r\nc.<\/td>\r\n0.001<\/td>\r\ng.<\/td>\r\n<\/tr>\r\n
\u22120.0001<\/td>\r\nd.<\/td>\r\n0.0001<\/td>\r\nh.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
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8.\u00a0<\/strong>Use the preceding two exercises to conjecture (guess) the value of the following limit: [latex]\\underset{x\\to 0}{\\lim}\\dfrac{\\sin ax}{x}[\/latex] for [latex]a[\/latex], a positive real value.<\/p>\r\n[reveal-answer q=\"fs-id1170572499871\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572499871\"]\r\n

[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin ax}{x}=a[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (9-14), set up a table of values to find the indicated limit. Round to eight digits.<\/p>\r\n\r\n

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9. [T]\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}\\dfrac{x^2-4}{x^2+x-6}[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/th>\r\n[latex]\\frac{x^2-4}{x^2+x-6}[\/latex]<\/th>\r\n[latex]x[\/latex]<\/th>\r\n[latex]\\frac{x^2-4}{x^2+x-6}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
1.9<\/td>\r\na.<\/td>\r\n2.1<\/td>\r\ne.<\/td>\r\n<\/tr>\r\n
1.99<\/td>\r\nb.<\/td>\r\n2.01<\/td>\r\nf.<\/td>\r\n<\/tr>\r\n
1.999<\/td>\r\nc.<\/td>\r\n2.001<\/td>\r\ng.<\/td>\r\n<\/tr>\r\n
1.9999<\/td>\r\nd.<\/td>\r\n2.0001<\/td>\r\nh.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
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10. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}(1-2x)[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/th>\r\n[latex]1-2x[\/latex]<\/th>\r\n[latex]x[\/latex]<\/th>\r\n[latex]1-2x[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
0.9<\/td>\r\na.<\/td>\r\n1.1<\/td>\r\ne.<\/td>\r\n<\/tr>\r\n
0.99<\/td>\r\nb.<\/td>\r\n1.01<\/td>\r\nf.<\/td>\r\n<\/tr>\r\n
0.999<\/td>\r\nc.<\/td>\r\n1.001<\/td>\r\ng.<\/td>\r\n<\/tr>\r\n
0.9999<\/td>\r\nd.<\/td>\r\n1.0001<\/td>\r\nh.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"fs-id1170571600021\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571600021\"]\r\n

a. \u22120.80000000; b. \u22120.98000000; c. \u22120.99800000; d. \u22120.99980000; e. \u22121.2000000; f. \u22121.0200000; g. \u22121.0020000; h. \u22121.0002000;<\/p>\r\n[latex]\\underset{x\\to 1}{\\lim}(1-2x)=-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\dfrac{5}{1-e^{1\/x}}[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/th>\r\n[latex]\\frac{5}{1-e^{1\/x}}[\/latex]<\/th>\r\n[latex]x[\/latex]<\/th>\r\n[latex]\\frac{5}{1-e^{1\/x}}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
\u22120.1<\/td>\r\na.<\/td>\r\n0.1<\/td>\r\ne.<\/td>\r\n<\/tr>\r\n
\u22120.01<\/td>\r\nb.<\/td>\r\n0.01<\/td>\r\nf.<\/td>\r\n<\/tr>\r\n
\u22120.001<\/td>\r\nc.<\/td>\r\n0.001<\/td>\r\ng.<\/td>\r\n<\/tr>\r\n
\u22120.0001<\/td>\r\nd.<\/td>\r\n0.0001<\/td>\r\nh.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
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12. [T]\u00a0<\/strong>[latex]\\underset{z\\to 0}{\\lim}\\dfrac{z-1}{z^2(z+3)}[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]z[\/latex]<\/th>\r\n[latex]\\frac{z-1}{z^2(z+3)}[\/latex]<\/th>\r\n[latex]z[\/latex]<\/th>\r\n[latex]\\frac{z-1}{z^2(z+3)}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
\u22120.1<\/td>\r\na.<\/td>\r\n0.1<\/td>\r\ne.<\/td>\r\n<\/tr>\r\n
\u22120.01<\/td>\r\nb.<\/td>\r\n0.01<\/td>\r\nf.<\/td>\r\n<\/tr>\r\n
\u22120.001<\/td>\r\nc.<\/td>\r\n0.001<\/td>\r\ng.<\/td>\r\n<\/tr>\r\n
\u22120.0001<\/td>\r\nd.<\/td>\r\n0.0001<\/td>\r\nh.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"fs-id1170572306112\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572306112\"]\r\n

a. \u221237.931934; b. \u22123377.9264; c. \u2212333,777.93; d. \u221233,337,778; e. \u221229.032258; f. \u22123289.0365; g. \u2212332,889.04; h. \u221233,328,889<\/p>\r\n[latex]\\underset{x\\to 0}{\\lim}\\frac{z-1}{z^2(z+3)}=\u2212\\infty [\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [T]\u00a0<\/strong>[latex]\\underset{t\\to 0^+}{\\lim}\\dfrac{\\cos t}{t}[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]t[\/latex]<\/th>\r\n[latex]\\frac{\\cos t}{t}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
0.1<\/td>\r\na.<\/td>\r\n<\/tr>\r\n
0.01<\/td>\r\nb.<\/td>\r\n<\/tr>\r\n
0.001<\/td>\r\nc.<\/td>\r\n<\/tr>\r\n
0.0001<\/td>\r\nd.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
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14. [T]\u00a0<\/strong>[latex]\\underset{x\\to 2^-}{\\lim}\\dfrac{1-\\frac{2}{x}}{x^2-4}[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/th>\r\n[latex]\\frac{1-\\frac{2}{x}}{x^2-4}[\/latex]<\/th>\r\n[latex]x[\/latex]<\/th>\r\n[latex]\\frac{1-\\frac{2}{x}}{x^2-4}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
1.9<\/td>\r\na.<\/td>\r\n2.1<\/td>\r\ne.<\/td>\r\n<\/tr>\r\n
1.99<\/td>\r\nb.<\/td>\r\n2.01<\/td>\r\nf.<\/td>\r\n<\/tr>\r\n
1.999<\/td>\r\nc.<\/td>\r\n2.001<\/td>\r\ng.<\/td>\r\n<\/tr>\r\n
1.9999<\/td>\r\nd.<\/td>\r\n2.0001<\/td>\r\nh.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"fs-id1170571610864\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571610864\"]\r\n

a. 0.13495277; b. 0.12594300; c. 0.12509381; d. 0.12500938; e. 0.11614402; f. 0.12406794; g. 0.12490631; h. 0.12499063;<\/p>\r\n[latex]\\underset{x\\to 2^-}{\\lim}\\frac{1-\\frac{2}{x}}{x^2-4}=0.1250=\\frac{1}{8}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (15-16), set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail?<\/p>\r\n\r\n

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15. [T]\u00a0<\/strong>[latex]\\underset{\\theta \\to 0^-}{\\lim}\\sin \\left(\\frac{\\pi }{\\theta }\\right)[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u03b8<\/em><\/th>\r\n[latex] \\sin (\\frac{\\pi }{\\theta })[\/latex]<\/th>\r\n\u03b8<\/em><\/th>\r\n[latex] \\sin (\\frac{\\pi }{\\theta })[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
\u22120.1<\/td>\r\na.<\/td>\r\n0.1<\/td>\r\ne.<\/td>\r\n<\/tr>\r\n
\u22120.01<\/td>\r\nb.<\/td>\r\n0.01<\/td>\r\nf.<\/td>\r\n<\/tr>\r\n
\u22120.001<\/td>\r\nc.<\/td>\r\n0.001<\/td>\r\ng.<\/td>\r\n<\/tr>\r\n
\u22120.0001<\/td>\r\nd.<\/td>\r\n0.0001<\/td>\r\nh.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
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16. [T]\u00a0<\/strong>[latex]\\underset{\\alpha \\to 0^+}{\\lim}\\frac{1}{\\alpha} \\cos \\left(\\frac{\\pi }{\\alpha }\\right)[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]a[\/latex]<\/th>\r\n[latex]\\frac{1}{\\alpha } \\cos (\\frac{\\pi }{\\alpha })[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
0.1<\/td>\r\na.<\/td>\r\n<\/tr>\r\n
0.01<\/td>\r\nb.<\/td>\r\n<\/tr>\r\n
0.001<\/td>\r\nc.<\/td>\r\n<\/tr>\r\n
0.0001<\/td>\r\nd.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"fs-id1170572243170\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572243170\"]\r\n

a. \u221210.00000; b. \u2212100.00000; c. \u22121000.0000; d. \u221210,000.000; Guess: [latex]\\underset{\\alpha \\to 0^+}{\\lim}\\frac{1}{\\alpha } \\cos (\\frac{\\pi }{\\alpha })=\\infty[\/latex], Actual: DNE<\/p>\r\n\"A<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (17-20), consider the graph of the function [latex]y=f(x)[\/latex] shown here. Which of the statements about [latex]y=f(x)[\/latex] are true and which are false? Explain why a statement is false.<\/p>\r\n\"A<\/span>\r\n

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17.\u00a0<\/strong>[latex]\\underset{x\\to 10^-}{\\lim}f(x)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>[latex]\\underset{x\\to -2^+}{\\lim}f(x)=3[\/latex]<\/p>\r\n\r\n

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False; [latex]\\underset{x\\to -2^+}{\\lim}f(x)=+\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0<\/strong>[latex]\\underset{x\\to -8^+}{\\lim}f(x)=f(-8)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]\\underset{x\\to 6}{\\lim}f(x)=5[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572548971\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572548971\"]\r\n

False; [latex]\\underset{x\\to 6}{\\lim}f(x)[\/latex] DNE since [latex]\\underset{x\\to 6^-}{\\lim}f(x)=2[\/latex] and [latex]\\underset{x\\to 6^+}{\\lim}f(x)=5[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (21-24), use the following graph of the function [latex]y=f(x)[\/latex] to find the values, if possible. Estimate when necessary.<\/p>\r\n\"A<\/span>\r\n

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21.\u00a0<\/strong>[latex]\\underset{x\\to 1^-}{\\lim}f(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n22.\u00a0<\/strong>[latex]\\underset{x\\to 1^+}{\\lim}f(x)[\/latex]\r\n\r\n[reveal-answer q=\"fs-id1170572540762\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572540762\"]\r\n

2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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23.\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

24.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex]<\/div>\r\n
25.\u00a0<\/strong>[latex]f(1)[\/latex]<\/div>\r\n

In the following exercises (26-29), use the graph of the function [latex]y=f(x)[\/latex] shown here to find the values, if possible. Estimate when necessary.<\/p>\r\n\"A<\/span>\r\n

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26.\u00a0<\/strong>[latex]\\underset{x\\to 0^-}{\\lim}f(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571563282\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571563282\"]\r\n

1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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27.\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}f(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}f(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571563366\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571563366\"]\r\n

DNE<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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29.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (30-35), use the graph of the function [latex]y=f(x)[\/latex] shown here to find the values, if possible. Estimate when necessary.<\/p>\r\n\"A<\/span>\r\n

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30.\u00a0<\/strong>[latex]\\underset{x\\to -2^-}{\\lim}f(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572624064\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572624064\"]\r\n

0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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31.\u00a0<\/strong>[latex]\\underset{x\\to -2^+}{\\lim}f(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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32.\u00a0<\/strong>[latex]\\underset{x\\to -2}{\\lim}f(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572624152\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572624152\"]\r\n

DNE<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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33.\u00a0<\/strong>[latex]\\underset{x\\to 2^-}{\\lim}f(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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34.\u00a0<\/strong>[latex]\\underset{x\\to 2^+}{\\lim}f(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572624239\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572624239\"]\r\n

2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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35.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (36-38), use the graph of the function [latex]y=g(x)[\/latex] shown here to find the values, if possible. Estimate when necessary.<\/p>\r\n\"A<\/span>\r\n

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36.\u00a0<\/strong>[latex]\\underset{x\\to 0^-}{\\lim}g(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572380992\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572380992\"]\r\n

3<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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37.\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}g(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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38.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}g(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572381076\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572381076\"]\r\n

DNE<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (39-41), use the graph of the function [latex]y=h(x)[\/latex] shown here to find the values, if possible. Estimate when necessary.<\/p>\r\n\"A<\/span>\r\n

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39.\u00a0<\/strong>[latex]\\underset{x\\to 0^-}{\\lim}h(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40.\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}h(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572372700\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572372700\"]\r\n

0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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41.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}h(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (42-46), use the graph of the function [latex]y=f(x)[\/latex] shown here to find the values, if possible. Estimate when necessary.<\/p>\r\n\"A<\/span>\r\n

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42.\u00a0<\/strong>[latex]\\underset{x\\to 0^-}{\\lim}f(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572267934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572267934\"]\r\n

\u22122<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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43.\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}f(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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44.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}f(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572268018\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572268018\"]\r\n

DNE<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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45.\u00a0<\/strong>[latex]\\underset{x\\to 1^-}{\\lim}f(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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46.\u00a0<\/strong>[latex]\\underset{x\\to 2^+}{\\lim}f(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572268099\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572268099\"]\r\n

0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (47-51), sketch the graph of a function with the given properties.<\/p>\r\n\r\n

\r\n\r\n47.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}f(x)=1, \\, \\underset{x\\to 4^-}{\\lim}f(x)=3[\/latex], [latex]\\underset{x\\to 4^+}{\\lim}f(x)=6, \\, f(4)[\/latex] is not defined.\r\n\r\n<\/div>\r\n
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48.\u00a0<\/strong>[latex]\\underset{x\\to -\\infty }{\\lim}f(x)=0, \\, \\underset{x\\to -1^-}{\\lim}f(x)=\u2212\\infty[\/latex], [latex]\\underset{x\\to -1^+}{\\lim}f(x)=\\infty, \\, \\underset{x\\to 0}{\\lim}f(x)=f(0), \\, f(0)=1, \\, \\underset{x\\to \\infty }{\\lim}f(x)=\u2212\\infty [\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572435005\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572435005\"]\r\n

Answers may vary.<\/p>\r\n\"A<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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49.\u00a0<\/strong>[latex]\\underset{x\\to -\\infty}{\\lim}f(x)=2, \\, \\underset{x\\to 3^-}{\\lim}f(x)=\u2212\\infty[\/latex], [latex]\\underset{x\\to 3^+}{\\lim}f(x)=\\infty, \\, \\underset{x\\to \\infty }{\\lim}f(x)=2, \\, f(0)=-\\frac{1}{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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50.\u00a0<\/strong>[latex]\\underset{x\\to -\\infty }{\\lim}f(x)=2, \\, \\underset{x\\to -2}{\\lim}f(x)=\u2212\\infty[\/latex],[latex]\\underset{x\\to \\infty }{\\lim}f(x)=2, \\, f(0)=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572552619\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572552619\"]\r\n

Answers may vary.<\/p>\r\n\"A<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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51.\u00a0<\/strong>[latex]\\underset{x\\to -\\infty }{\\lim}f(x)=0, \\, \\underset{x\\to -1^-}{\\lim}f(x)=\\infty, \\, \\underset{x\\to -1^+}{\\lim}f(x)=\u2212\\infty, \\, f(0)=-1, \\, \\underset{x\\to 1^-}{\\lim}f(x)=\u2212\\infty, \\, \\underset{x\\to 1^+}{\\lim}f(x)=\\infty, \\, \\underset{x\\to \\infty }{\\lim}f(x)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n\r\n52.\u00a0<\/strong>Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, [latex]x[\/latex], is shown here. We are mainly interested in the location of the front of the shock, labeled [latex]x_{SF}[\/latex] in the diagram.\r\n\r\n\"A\r\n\r\na. Evaluate\u00a0[latex]\\underset{x\\to x_{SF}^+}{\\lim}\\rho(x)[\/latex]\r\n\r\nb. Evaluate\u00a0[latex]\\underset{x\\to x_{SF}^-}{\\lim}\\rho(x)[\/latex]\r\n\r\nc. Evaluate\u00a0[latex]\\underset{x\\to x_{SF}}{\\lim}\\rho(x)[\/latex].\u00a0Explain the physical meanings behind your answers.\r\n\r\n[reveal-answer q=\"54988033\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"54988033\"]\r\n\r\na.\u00a0[latex]\\rho_{2}[\/latex]\r\n\r\nb. [latex]\\rho_{1}[\/latex]\r\n\r\nc. DNE unless\u00a0[latex]\\rho_{1}=\\rho_{2}[\/latex]<\/span>\r\n\r\nAs you approach [latex]x_{SF}[\/latex]\u00a0from the right, you are in the high-density area of the shock. When you approach from the left, you have not experienced the \u201cshock\u201d yet and are at a lower density.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
\r\n\r\n53.\u00a0<\/strong>A track coach uses a camera with a fast shutter to estimate the position of a runner with respect to time. A table of the values of position of the athlete versus time is given here, where [latex]x[\/latex] is the position in meters of the runner and [latex]t[\/latex] is time in seconds. What is [latex]\\underset{t\\to 2}{\\lim}x(t)[\/latex]? What does it mean physically?\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]t[\/latex] (sec)<\/th>\r\n[latex]x[\/latex] (m)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
1.75<\/td>\r\n4.5<\/td>\r\n<\/tr>\r\n
1.95<\/td>\r\n6.1<\/td>\r\n<\/tr>\r\n
1.99<\/td>\r\n6.42<\/td>\r\n<\/tr>\r\n
2.01<\/td>\r\n6.58<\/td>\r\n<\/tr>\r\n
2.05<\/td>\r\n6.9<\/td>\r\n<\/tr>\r\n
2.25<\/td>\r\n8.5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>","rendered":"
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For the following exercises (1-2), consider the function [latex]f(x)=\\dfrac{x^2-1}{|x-1|}[\/latex].<\/p>\n

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1. [T]<\/strong> Complete the following table for the function. Round your solutions to four decimal places.<\/p>\n\n\n\n\n\n\n\n\n
[latex]x[\/latex]<\/th>\n[latex]f(x)[\/latex]<\/th>\n<\/th>\n[latex]x[\/latex]<\/th>\n[latex]f(x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
0.9<\/td>\na.<\/td>\n<\/td>\n1.1<\/td>\ne.<\/td>\n<\/tr>\n
0.99<\/td>\nb.<\/td>\n<\/td>\n1.01<\/td>\nf.<\/td>\n<\/tr>\n
0.999<\/td>\nc.<\/td>\n<\/td>\n1.001<\/td>\ng.<\/td>\n<\/tr>\n
0.9999<\/td>\nd.<\/td>\n<\/td>\n1.0001<\/td>\nh.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>What do your results in the preceding exercise indicate about the two-sided limit [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex]? Explain your response.<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex] does not exist because [latex]\\underset{x\\to 1^-}{\\lim}f(x)=-2 \\ne \\underset{x\\to 1^+}{\\lim}f(x)=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (3-5), consider the function [latex]f(x)=(1+x)^{1\/x}[\/latex].<\/p>\n

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3. [T]<\/strong> Make a table showing the values of [latex]f[\/latex] for [latex]x=-0.01, \\, -0.001, \\, -0.0001, \\, -0.00001[\/latex] and for [latex]x=0.01, \\, 0.001, \\, 0.0001, \\, 0.00001[\/latex]. Round your solutions to five decimal places.<\/p>\n\n\n\n\n\n\n\n\n
[latex]x[\/latex]<\/th>\n[latex]f(x)[\/latex]<\/th>\n<\/th>\n[latex]x[\/latex]<\/th>\n[latex]f(x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
\u22120.01<\/td>\na.<\/td>\n<\/td>\n0.01<\/td>\ne.<\/td>\n<\/tr>\n
\u22120.001<\/td>\nb.<\/td>\n<\/td>\n0.001<\/td>\nf.<\/td>\n<\/tr>\n
\u22120.0001<\/td>\nc.<\/td>\n<\/td>\n0.0001<\/td>\ng.<\/td>\n<\/tr>\n
\u22120.00001<\/td>\nd.<\/td>\n<\/td>\n0.00001<\/td>\nh.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
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4.\u00a0<\/strong>What does the table of values in the preceding exercise indicate about the function [latex]f(x)=(1+x)^{1\/x}[\/latex]?<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\underset{x\\to 0}{\\lim}(1+x)^{1\/x}=2.7183[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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5.\u00a0<\/strong>To which mathematical constant does the limit in the preceding exercise appear to be getting closer?<\/p>\n<\/div>\n<\/div>\n

In the following exercises (6-8), use the given values of [latex]x[\/latex] to set up a table to evaluate the limits. Round your solutions to eight decimal places.<\/p>\n

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6. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\dfrac{\\sin 2x}{x}; \\, x = \\pm 0.1, \\, \\pm 0.01, \\, \\pm 0.001, \\, \\pm 0.0001[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
[latex]x[\/latex]<\/th>\n[latex]\\frac{\\sin 2x}{x}[\/latex]<\/th>\n[latex]x[\/latex]<\/th>\n[latex]\\frac{\\sin 2x}{x}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
\u22120.1<\/td>\na.<\/td>\n0.1<\/td>\ne.<\/td>\n<\/tr>\n
\u22120.01<\/td>\nb.<\/td>\n0.01<\/td>\nf.<\/td>\n<\/tr>\n
\u22120.001<\/td>\nc.<\/td>\n0.001<\/td>\ng.<\/td>\n<\/tr>\n
\u22120.0001<\/td>\nd.<\/td>\n0.0001<\/td>\nh.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Show Solution<\/span><\/p>\n
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a. 1.98669331; b. 1.99986667; c. 1.99999867; d. 1.99999999; e. 1.98669331; f. 1.99986667; g. 1.99999867; h. 1.99999999; [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin 2x}{x}=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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7. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\dfrac{\\sin 3x}{x}; \\, x = \\pm 0.1, \\, \\pm 0.01, \\, \\pm 0.001, \\, \\pm 0.0001[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
X<\/em><\/th>\n[latex]\\frac{\\sin 3x}{x}[\/latex]<\/th>\n[latex]x[\/latex]<\/th>\n[latex]\\frac{\\sin 3x}{x}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
\u22120.1<\/td>\na.<\/td>\n0.1<\/td>\ne.<\/td>\n<\/tr>\n
\u22120.01<\/td>\nb.<\/td>\n0.01<\/td>\nf.<\/td>\n<\/tr>\n
\u22120.001<\/td>\nc.<\/td>\n0.001<\/td>\ng.<\/td>\n<\/tr>\n
\u22120.0001<\/td>\nd.<\/td>\n0.0001<\/td>\nh.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
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8.\u00a0<\/strong>Use the preceding two exercises to conjecture (guess) the value of the following limit: [latex]\\underset{x\\to 0}{\\lim}\\dfrac{\\sin ax}{x}[\/latex] for [latex]a[\/latex], a positive real value.<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin ax}{x}=a[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises (9-14), set up a table of values to find the indicated limit. Round to eight digits.<\/p>\n

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9. [T]\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}\\dfrac{x^2-4}{x^2+x-6}[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
[latex]x[\/latex]<\/th>\n[latex]\\frac{x^2-4}{x^2+x-6}[\/latex]<\/th>\n[latex]x[\/latex]<\/th>\n[latex]\\frac{x^2-4}{x^2+x-6}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
1.9<\/td>\na.<\/td>\n2.1<\/td>\ne.<\/td>\n<\/tr>\n
1.99<\/td>\nb.<\/td>\n2.01<\/td>\nf.<\/td>\n<\/tr>\n
1.999<\/td>\nc.<\/td>\n2.001<\/td>\ng.<\/td>\n<\/tr>\n
1.9999<\/td>\nd.<\/td>\n2.0001<\/td>\nh.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
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10. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}(1-2x)[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
[latex]x[\/latex]<\/th>\n[latex]1-2x[\/latex]<\/th>\n[latex]x[\/latex]<\/th>\n[latex]1-2x[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
0.9<\/td>\na.<\/td>\n1.1<\/td>\ne.<\/td>\n<\/tr>\n
0.99<\/td>\nb.<\/td>\n1.01<\/td>\nf.<\/td>\n<\/tr>\n
0.999<\/td>\nc.<\/td>\n1.001<\/td>\ng.<\/td>\n<\/tr>\n
0.9999<\/td>\nd.<\/td>\n1.0001<\/td>\nh.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Show Solution<\/span><\/p>\n
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a. \u22120.80000000; b. \u22120.98000000; c. \u22120.99800000; d. \u22120.99980000; e. \u22121.2000000; f. \u22121.0200000; g. \u22121.0020000; h. \u22121.0002000;<\/p>\n

[latex]\\underset{x\\to 1}{\\lim}(1-2x)=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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11. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\dfrac{5}{1-e^{1\/x}}[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
[latex]x[\/latex]<\/th>\n[latex]\\frac{5}{1-e^{1\/x}}[\/latex]<\/th>\n[latex]x[\/latex]<\/th>\n[latex]\\frac{5}{1-e^{1\/x}}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
\u22120.1<\/td>\na.<\/td>\n0.1<\/td>\ne.<\/td>\n<\/tr>\n
\u22120.01<\/td>\nb.<\/td>\n0.01<\/td>\nf.<\/td>\n<\/tr>\n
\u22120.001<\/td>\nc.<\/td>\n0.001<\/td>\ng.<\/td>\n<\/tr>\n
\u22120.0001<\/td>\nd.<\/td>\n0.0001<\/td>\nh.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
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12. [T]\u00a0<\/strong>[latex]\\underset{z\\to 0}{\\lim}\\dfrac{z-1}{z^2(z+3)}[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
[latex]z[\/latex]<\/th>\n[latex]\\frac{z-1}{z^2(z+3)}[\/latex]<\/th>\n[latex]z[\/latex]<\/th>\n[latex]\\frac{z-1}{z^2(z+3)}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
\u22120.1<\/td>\na.<\/td>\n0.1<\/td>\ne.<\/td>\n<\/tr>\n
\u22120.01<\/td>\nb.<\/td>\n0.01<\/td>\nf.<\/td>\n<\/tr>\n
\u22120.001<\/td>\nc.<\/td>\n0.001<\/td>\ng.<\/td>\n<\/tr>\n
\u22120.0001<\/td>\nd.<\/td>\n0.0001<\/td>\nh.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Show Solution<\/span><\/p>\n
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a. \u221237.931934; b. \u22123377.9264; c. \u2212333,777.93; d. \u221233,337,778; e. \u221229.032258; f. \u22123289.0365; g. \u2212332,889.04; h. \u221233,328,889<\/p>\n

[latex]\\underset{x\\to 0}{\\lim}\\frac{z-1}{z^2(z+3)}=\u2212\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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13. [T]\u00a0<\/strong>[latex]\\underset{t\\to 0^+}{\\lim}\\dfrac{\\cos t}{t}[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
[latex]t[\/latex]<\/th>\n[latex]\\frac{\\cos t}{t}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
0.1<\/td>\na.<\/td>\n<\/tr>\n
0.01<\/td>\nb.<\/td>\n<\/tr>\n
0.001<\/td>\nc.<\/td>\n<\/tr>\n
0.0001<\/td>\nd.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
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14. [T]\u00a0<\/strong>[latex]\\underset{x\\to 2^-}{\\lim}\\dfrac{1-\\frac{2}{x}}{x^2-4}[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
[latex]x[\/latex]<\/th>\n[latex]\\frac{1-\\frac{2}{x}}{x^2-4}[\/latex]<\/th>\n[latex]x[\/latex]<\/th>\n[latex]\\frac{1-\\frac{2}{x}}{x^2-4}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
1.9<\/td>\na.<\/td>\n2.1<\/td>\ne.<\/td>\n<\/tr>\n
1.99<\/td>\nb.<\/td>\n2.01<\/td>\nf.<\/td>\n<\/tr>\n
1.999<\/td>\nc.<\/td>\n2.001<\/td>\ng.<\/td>\n<\/tr>\n
1.9999<\/td>\nd.<\/td>\n2.0001<\/td>\nh.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Show Solution<\/span><\/p>\n
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a. 0.13495277; b. 0.12594300; c. 0.12509381; d. 0.12500938; e. 0.11614402; f. 0.12406794; g. 0.12490631; h. 0.12499063;<\/p>\n

[latex]\\underset{x\\to 2^-}{\\lim}\\frac{1-\\frac{2}{x}}{x^2-4}=0.1250=\\frac{1}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises (15-16), set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail?<\/p>\n

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15. [T]\u00a0<\/strong>[latex]\\underset{\\theta \\to 0^-}{\\lim}\\sin \\left(\\frac{\\pi }{\\theta }\\right)[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
\u03b8<\/em><\/th>\n[latex]\\sin (\\frac{\\pi }{\\theta })[\/latex]<\/th>\n\u03b8<\/em><\/th>\n[latex]\\sin (\\frac{\\pi }{\\theta })[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
\u22120.1<\/td>\na.<\/td>\n0.1<\/td>\ne.<\/td>\n<\/tr>\n
\u22120.01<\/td>\nb.<\/td>\n0.01<\/td>\nf.<\/td>\n<\/tr>\n
\u22120.001<\/td>\nc.<\/td>\n0.001<\/td>\ng.<\/td>\n<\/tr>\n
\u22120.0001<\/td>\nd.<\/td>\n0.0001<\/td>\nh.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
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16. [T]\u00a0<\/strong>[latex]\\underset{\\alpha \\to 0^+}{\\lim}\\frac{1}{\\alpha} \\cos \\left(\\frac{\\pi }{\\alpha }\\right)[\/latex]<\/p>\n\n\n\n\n\n\n\n\n
[latex]a[\/latex]<\/th>\n[latex]\\frac{1}{\\alpha } \\cos (\\frac{\\pi }{\\alpha })[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
0.1<\/td>\na.<\/td>\n<\/tr>\n
0.01<\/td>\nb.<\/td>\n<\/tr>\n
0.001<\/td>\nc.<\/td>\n<\/tr>\n
0.0001<\/td>\nd.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Show Solution<\/span><\/p>\n
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a. \u221210.00000; b. \u2212100.00000; c. \u22121000.0000; d. \u221210,000.000; Guess: [latex]\\underset{\\alpha \\to 0^+}{\\lim}\\frac{1}{\\alpha } \\cos (\\frac{\\pi }{\\alpha })=\\infty[\/latex], Actual: DNE<\/p>\n

\"A<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises (17-20), consider the graph of the function [latex]y=f(x)[\/latex] shown here. Which of the statements about [latex]y=f(x)[\/latex] are true and which are false? Explain why a statement is false.<\/p>\n

\"A<\/span><\/p>\n

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17.\u00a0<\/strong>[latex]\\underset{x\\to 10^-}{\\lim}f(x)=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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18.\u00a0<\/strong>[latex]\\underset{x\\to -2^+}{\\lim}f(x)=3[\/latex]<\/p>\n

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Show Solution<\/span><\/p>\n
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False; [latex]\\underset{x\\to -2^+}{\\lim}f(x)=+\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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19.\u00a0<\/strong>[latex]\\underset{x\\to -8^+}{\\lim}f(x)=f(-8)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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20.\u00a0<\/strong>[latex]\\underset{x\\to 6}{\\lim}f(x)=5[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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False; [latex]\\underset{x\\to 6}{\\lim}f(x)[\/latex] DNE since [latex]\\underset{x\\to 6^-}{\\lim}f(x)=2[\/latex] and [latex]\\underset{x\\to 6^+}{\\lim}f(x)=5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises (21-24), use the following graph of the function [latex]y=f(x)[\/latex] to find the values, if possible. Estimate when necessary.<\/p>\n

\"A<\/span><\/p>\n

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21.\u00a0<\/strong>[latex]\\underset{x\\to 1^-}{\\lim}f(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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22.\u00a0<\/strong>[latex]\\underset{x\\to 1^+}{\\lim}f(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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23.\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

24.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex]<\/div>\n
25.\u00a0<\/strong>[latex]f(1)[\/latex]<\/div>\n

In the following exercises (26-29), use the graph of the function [latex]y=f(x)[\/latex] shown here to find the values, if possible. Estimate when necessary.<\/p>\n

\"A<\/span><\/p>\n

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26.\u00a0<\/strong>[latex]\\underset{x\\to 0^-}{\\lim}f(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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27.\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}f(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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28.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}f(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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DNE<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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29.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

In the following exercises (30-35), use the graph of the function [latex]y=f(x)[\/latex] shown here to find the values, if possible. Estimate when necessary.<\/p>\n

\"A<\/span><\/p>\n

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30.\u00a0<\/strong>[latex]\\underset{x\\to -2^-}{\\lim}f(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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31.\u00a0<\/strong>[latex]\\underset{x\\to -2^+}{\\lim}f(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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32.\u00a0<\/strong>[latex]\\underset{x\\to -2}{\\lim}f(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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DNE<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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33.\u00a0<\/strong>[latex]\\underset{x\\to 2^-}{\\lim}f(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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34.\u00a0<\/strong>[latex]\\underset{x\\to 2^+}{\\lim}f(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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35.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

In the following exercises (36-38), use the graph of the function [latex]y=g(x)[\/latex] shown here to find the values, if possible. Estimate when necessary.<\/p>\n

\"A<\/span><\/p>\n

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36.\u00a0<\/strong>[latex]\\underset{x\\to 0^-}{\\lim}g(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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37.\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}g(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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38.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}g(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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DNE<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises (39-41), use the graph of the function [latex]y=h(x)[\/latex] shown here to find the values, if possible. Estimate when necessary.<\/p>\n

\"A<\/span><\/p>\n

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39.\u00a0<\/strong>[latex]\\underset{x\\to 0^-}{\\lim}h(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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40.\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}h(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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41.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}h(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

In the following exercises (42-46), use the graph of the function [latex]y=f(x)[\/latex] shown here to find the values, if possible. Estimate when necessary.<\/p>\n

\"A<\/span><\/p>\n

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42.\u00a0<\/strong>[latex]\\underset{x\\to 0^-}{\\lim}f(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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\u22122<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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43.\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}f(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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44.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}f(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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DNE<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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45.\u00a0<\/strong>[latex]\\underset{x\\to 1^-}{\\lim}f(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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46.\u00a0<\/strong>[latex]\\underset{x\\to 2^+}{\\lim}f(x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises (47-51), sketch the graph of a function with the given properties.<\/p>\n

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47.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}f(x)=1, \\, \\underset{x\\to 4^-}{\\lim}f(x)=3[\/latex], [latex]\\underset{x\\to 4^+}{\\lim}f(x)=6, \\, f(4)[\/latex] is not defined.<\/p>\n<\/div>\n

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48.\u00a0<\/strong>[latex]\\underset{x\\to -\\infty }{\\lim}f(x)=0, \\, \\underset{x\\to -1^-}{\\lim}f(x)=\u2212\\infty[\/latex], [latex]\\underset{x\\to -1^+}{\\lim}f(x)=\\infty, \\, \\underset{x\\to 0}{\\lim}f(x)=f(0), \\, f(0)=1, \\, \\underset{x\\to \\infty }{\\lim}f(x)=\u2212\\infty[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Answers may vary.<\/p>\n

\"A<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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49.\u00a0<\/strong>[latex]\\underset{x\\to -\\infty}{\\lim}f(x)=2, \\, \\underset{x\\to 3^-}{\\lim}f(x)=\u2212\\infty[\/latex], [latex]\\underset{x\\to 3^+}{\\lim}f(x)=\\infty, \\, \\underset{x\\to \\infty }{\\lim}f(x)=2, \\, f(0)=-\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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50.\u00a0<\/strong>[latex]\\underset{x\\to -\\infty }{\\lim}f(x)=2, \\, \\underset{x\\to -2}{\\lim}f(x)=\u2212\\infty[\/latex],[latex]\\underset{x\\to \\infty }{\\lim}f(x)=2, \\, f(0)=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Answers may vary.<\/p>\n

\"A<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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51.\u00a0<\/strong>[latex]\\underset{x\\to -\\infty }{\\lim}f(x)=0, \\, \\underset{x\\to -1^-}{\\lim}f(x)=\\infty, \\, \\underset{x\\to -1^+}{\\lim}f(x)=\u2212\\infty, \\, f(0)=-1, \\, \\underset{x\\to 1^-}{\\lim}f(x)=\u2212\\infty, \\, \\underset{x\\to 1^+}{\\lim}f(x)=\\infty, \\, \\underset{x\\to \\infty }{\\lim}f(x)=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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52.\u00a0<\/strong>Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, [latex]x[\/latex], is shown here. We are mainly interested in the location of the front of the shock, labeled [latex]x_{SF}[\/latex] in the diagram.<\/p>\n

\"A<\/p>\n

a. Evaluate\u00a0[latex]\\underset{x\\to x_{SF}^+}{\\lim}\\rho(x)[\/latex]<\/p>\n

b. Evaluate\u00a0[latex]\\underset{x\\to x_{SF}^-}{\\lim}\\rho(x)[\/latex]<\/p>\n

c. Evaluate\u00a0[latex]\\underset{x\\to x_{SF}}{\\lim}\\rho(x)[\/latex].\u00a0Explain the physical meanings behind your answers.<\/p>\n

Show Solution<\/span><\/p>\n
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a.\u00a0[latex]\\rho_{2}[\/latex]<\/p>\n

b. [latex]\\rho_{1}[\/latex]<\/p>\n

c. DNE unless\u00a0[latex]\\rho_{1}=\\rho_{2}[\/latex]<\/span><\/p>\n

As you approach [latex]x_{SF}[\/latex]\u00a0from the right, you are in the high-density area of the shock. When you approach from the left, you have not experienced the \u201cshock\u201d yet and are at a lower density.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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53.\u00a0<\/strong>A track coach uses a camera with a fast shutter to estimate the position of a runner with respect to time. A table of the values of position of the athlete versus time is given here, where [latex]x[\/latex] is the position in meters of the runner and [latex]t[\/latex] is time in seconds. What is [latex]\\underset{t\\to 2}{\\lim}x(t)[\/latex]? What does it mean physically?<\/p>\n\n\n\n\n\n\n\n\n\n\n
[latex]t[\/latex] (sec)<\/th>\n[latex]x[\/latex] (m)<\/th>\n<\/tr>\n<\/thead>\n
1.75<\/td>\n4.5<\/td>\n<\/tr>\n
1.95<\/td>\n6.1<\/td>\n<\/tr>\n
1.99<\/td>\n6.42<\/td>\n<\/tr>\n
2.01<\/td>\n6.58<\/td>\n<\/tr>\n
2.05<\/td>\n6.9<\/td>\n<\/tr>\n
2.25<\/td>\n8.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n\n\t\t\t
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Candela Citations<\/h3>\n\t\t\t\t\t
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