{"id":459,"date":"2021-02-04T15:28:10","date_gmt":"2021-02-04T15:28:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=459"},"modified":"2021-03-30T21:28:30","modified_gmt":"2021-03-30T21:28:30","slug":"problem-set-the-precise-definition-of-a-limit","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-the-precise-definition-of-a-limit\/","title":{"raw":"Problem Set: The Precise Definition of a Limit","rendered":"Problem Set: The Precise Definition of a Limit"},"content":{"raw":"

In the following exercises (1-4), write the appropriate [latex]\\varepsilon[\/latex]-[latex]\\delta[\/latex] definition for each of the given statements.<\/p>\r\n\r\n

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

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2.\u00a0<\/strong>[latex]\\underset{t\\to b}{\\lim}g(t)=M[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571613444\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571613444\"]\r\n

For every [latex]\\varepsilon >0[\/latex], there exists a [latex]\\delta >0[\/latex] so that if [latex]0<|t-b|<\\delta[\/latex], then [latex]|g(t)-M|<\\varepsilon[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

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3.\u00a0<\/strong>[latex]\\underset{x\\to c}{\\lim}h(x)=L[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>[latex]\\underset{x\\to a}{\\lim}\\phi(x)=A[\/latex]<\/p>\r\n\r\n

\r\n\r\n[reveal-answer q=\"8722563\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"8722563\"]\r\n

For every [latex]\\varepsilon >0[\/latex], there exists a [latex]\\delta >0[\/latex] so that if [latex]0<|x-a|<\\delta[\/latex], then [latex]|\\phi(x)-A|<\\varepsilon[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 2}{\\lim}f(x)=2[\/latex]. In the following exercises (5-6), determine a value of [latex]\\delta >0[\/latex] that satisfies each statement.<\/p>\r\n\"A<\/span>\r\n

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5.\u00a0<\/strong>If [latex]0<|x-2|<\\delta[\/latex], then [latex]|f(x)-2|<1[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>If [latex]0<|x-2|<\\delta[\/latex], then [latex]|f(x)-2|<0.5[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170572130429\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572130429\"]\r\n

[latex]\\delta \\le 0.25[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 3}{\\lim}f(x)=-1[\/latex]. In the following exercises (7-8), determine a value of [latex]\\delta >0[\/latex] that satisfies each statement.<\/p>\r\n\"A<\/span>\r\n

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7.\u00a0<\/strong>If [latex]0<|x-3|<\\delta[\/latex], then [latex]|f(x)+1|<1[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>If [latex]0<|x-3|<\\delta[\/latex], then [latex]|f(x)+1|<2[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170571609280\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571609280\"]\r\n

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

The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 3}{\\lim}f(x)=2[\/latex]. In the following exercises (9-10), for each value of [latex]\\varepsilon[\/latex], find a value of [latex]\\delta >0[\/latex] such that the precise definition of limit holds true.<\/p>\r\n\"A<\/span>\r\n

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9.\u00a0<\/strong>[latex]\\varepsilon =1.5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>[latex]\\varepsilon =3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572618120\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572618120\"]\r\n

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

In the following exercises (11-12), use a graphing calculator to find a number [latex]\\delta[\/latex] such that the statements hold true.<\/p>\r\n\r\n

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11. [T]\u00a0<\/strong>[latex]|\\sin (2x)-\\frac{1}{2}|<0.1[\/latex], whenever [latex]|x-\\frac{\\pi}{12}|<\\delta[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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12. [T]\u00a0<\/strong>[latex]|\\sqrt{x-4}-2|<0.1[\/latex], whenever [latex]|x-8|<\\delta [\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572551787\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572551787\"]\r\n

[latex]\\delta <0.3900[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (13-17), use the precise definition of limit to prove the given limits.<\/p>\r\n\r\n

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\r\n\r\n13.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}(5x+8)=18[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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14.\u00a0<\/strong>[latex]\\underset{x\\to 3}{\\lim}\\dfrac{x^2-9}{x-3}=6[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572558414\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572558414\"]\r\n

Let [latex]\\delta =\\varepsilon[\/latex]. If [latex]0<|x-3|<\\varepsilon[\/latex], then [latex]|x+3-6|=|x-3|<\\varepsilon[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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15.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}\\dfrac{2x^2-3x-2}{x-2}=5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x^4=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571600697\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571600697\"]\r\n

Let [latex]\\delta =\\sqrt[4]{\\varepsilon}[\/latex]. If [latex]0<|x|<\\sqrt[4]{\\varepsilon}[\/latex], then [latex]|x^4|=x^4<\\varepsilon[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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17.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}(x^2+2x)=8[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (18-20), use the precise definition of limit to prove the given one-sided limits.<\/p>\r\n\r\n

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18.\u00a0<\/strong>[latex]\\underset{x\\to 5^-}{\\lim}\\sqrt{5-x}=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572229785\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572229785\"]\r\n

Let [latex]\\delta =\\varepsilon^2[\/latex]. If [latex]5-\\varepsilon^2<x<5[\/latex], then [latex]|\\sqrt{5-x}|=\\sqrt{5-x}<\\varepsilon[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}f(x)=-2[\/latex], where [latex]f(x)=\\begin{cases} 8x-3 & \\text{ if } \\, x<0 \\\\ 4x-2 & \\text{ if } \\, x \\ge 0 \\end{cases}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]\\underset{x\\to 1^-}{\\lim}f(x)=3[\/latex], where [latex]f(x)=\\begin{cases} 5x-2 & \\text{ if } \\, x < 1 \\\\ 7x-1 & \\text{ if } x \\ge 1 \\end{cases}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572184209\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572184209\"]\r\n

Let [latex]\\delta =\\varepsilon\/5[\/latex]. If [latex]1-\\varepsilon\/5<x<1[\/latex], then [latex]|f(x)-3|=5x-5<\\varepsilon[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

In the following exercises (21-23), use the precise definition of limit to prove the given infinite limits.<\/p>\r\n\r\n

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21.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\dfrac{1}{x^2}=\\infty [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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22.\u00a0<\/strong>[latex]\\underset{x\\to -1}{\\lim}\\dfrac{3}{(x+1)^2}=\\infty [\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572233918\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572233918\"]\r\n

Let [latex]\\delta =\\sqrt{\\frac{3}{N}}[\/latex]. If [latex]0<|x+1|<\\sqrt{\\frac{3}{N}}[\/latex], then [latex]f(x)=\\frac{3}{(x+1)^2}>N[\/latex].<\/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 2}{\\lim}-\\dfrac{1}{(x-2)^2}=\u2212\\infty [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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24.\u00a0<\/strong>An engineer is using a machine to cut a flat square of Aerogel of area 144 cm2<\/sup>. If there is a maximum error tolerance in the area of 8 cm2<\/sup>, how accurately must the engineer cut on the side, assuming all sides have the same length? How do these numbers relate to [latex]\\delta, \\, \\varepsilon, \\, a[\/latex], and [latex]L[\/latex]?<\/p>\r\n[reveal-answer q=\"fs-id1170571652932\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571652932\"]\r\n

0.033 cm, [latex]\\varepsilon =8, \\, \\delta =0.33, \\, a=12, \\, L=144[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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25.\u00a0<\/strong>Use the precise definition of limit to prove that the following limit does not exist: [latex]\\underset{x\\to 1}{\\lim}\\dfrac{|x-1|}{x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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26.\u00a0<\/strong>Using precise definitions of limits, prove that [latex]\\underset{x\\to 0}{\\lim}f(x)[\/latex] does not exist, given that [latex]f(x)[\/latex] is the ceiling function.<\/p>\r\n[reveal-answer q=\"67334902\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"67334902\"]\r\n\r\nTry any [latex]\\delta <1[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572626632\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572626632\"]\r\n

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

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27.\u00a0<\/strong>Using precise definitions of limits, prove that [latex]\\underset{x\\to 0}{\\lim}f(x)[\/latex] does not exist: [latex]f(x)=\\begin{cases} 1 & \\text{ if } \\, x \\, \\text{is rational} \\\\ 0 & \\text{ if } \\, x \\, \\text{is irrational} \\end{cases}[\/latex]<\/p>\r\n[reveal-answer q=\"94558722\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"94558722\"]\r\n\r\nThink about how you can always choose a rational number [latex]0<r<d[\/latex], but [latex]|f(r)-0|=1[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.\u00a0<\/strong>Using precise definitions of limits, determine [latex]\\underset{x\\to 0}{\\lim}f(x)[\/latex] for [latex]f(x)=\\begin{cases} x & \\text{ if } \\, x \\, \\text{is rational} \\\\ 0 & \\text{ if } \\, x \\, \\text{is irrational} \\end{cases}[\/latex]<\/p>\r\n[reveal-answer q=\"9093765\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"9093765\"]\r\n\r\nBreak into two cases, [latex]x[\/latex] rational and [latex]x[\/latex] irrational.\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170571661065\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571661065\"]\r\n

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

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29.\u00a0<\/strong>Using the function from the previous exercise, use the precise definition of limits to show that [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] does not exist for [latex]a\\ne 0[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (30-32), suppose that [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=M[\/latex] both exist. Use the precise definition of limits to prove the following limit laws:<\/p>\r\n\r\n

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30.\u00a0<\/strong>[latex]\\underset{x\\to a}{\\lim}(f(x)-g(x))=L-M[\/latex]<\/p>\r\n\r\n

\r\n\r\n[reveal-answer q=\"fs-id1170572293467\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572293467\"]\r\n

[latex]f(x)-g(x)=f(x)+(-1)g(x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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31.\u00a0<\/strong>[latex]\\underset{x\\to a}{\\lim}[cf(x)]=cL[\/latex] for any real constant [latex]c[\/latex]<\/p>\r\n[reveal-answer q=\"7115520\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"7115520\"]\r\n\r\nConsider two cases: [latex]c=0[\/latex] and [latex]c\\ne 0[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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32.\u00a0<\/strong>[latex]\\underset{x\\to a}{\\lim}[f(x)g(x)]=LM[\/latex].<\/p>\r\n[reveal-answer q=\"88552271\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"88552271\"]\r\n\r\n[latex]|f(x)g(x)-LM|=|f(x)g(x)-f(x)M+f(x)M-LM|\\le |f(x)||g(x)-M|+|M||f(x)-L|[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572565337\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572565337\"]\r\n

Answers may vary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

In the following exercises (1-4), write the appropriate [latex]\\varepsilon[\/latex]–[latex]\\delta[\/latex] definition for each of the given statements.<\/p>\n

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

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2.\u00a0<\/strong>[latex]\\underset{t\\to b}{\\lim}g(t)=M[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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For every [latex]\\varepsilon >0[\/latex], there exists a [latex]\\delta >0[\/latex] so that if [latex]0<|t-b|<\\delta[\/latex], then [latex]|g(t)-M|<\\varepsilon[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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3.\u00a0<\/strong>[latex]\\underset{x\\to c}{\\lim}h(x)=L[\/latex]<\/p>\n<\/div>\n<\/div>\n

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4.\u00a0<\/strong>[latex]\\underset{x\\to a}{\\lim}\\phi(x)=A[\/latex]<\/p>\n

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Show Solution<\/span><\/p>\n
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For every [latex]\\varepsilon >0[\/latex], there exists a [latex]\\delta >0[\/latex] so that if [latex]0<|x-a|<\\delta[\/latex], then [latex]|\\phi(x)-A|<\\varepsilon[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 2}{\\lim}f(x)=2[\/latex]. In the following exercises (5-6), determine a value of [latex]\\delta >0[\/latex] that satisfies each statement.<\/p>\n

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

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5.\u00a0<\/strong>If [latex]0<|x-2|<\\delta[\/latex], then [latex]|f(x)-2|<1[\/latex].<\/p>\n<\/div>\n<\/div>\n

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6.\u00a0<\/strong>If [latex]0<|x-2|<\\delta[\/latex], then [latex]|f(x)-2|<0.5[\/latex].<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\delta \\le 0.25[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 3}{\\lim}f(x)=-1[\/latex]. In the following exercises (7-8), determine a value of [latex]\\delta >0[\/latex] that satisfies each statement.<\/p>\n

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

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7.\u00a0<\/strong>If [latex]0<|x-3|<\\delta[\/latex], then [latex]|f(x)+1|<1[\/latex].<\/p>\n<\/div>\n<\/div>\n

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8.\u00a0<\/strong>If [latex]0<|x-3|<\\delta[\/latex], then [latex]|f(x)+1|<2[\/latex].<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\delta \\le 2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 3}{\\lim}f(x)=2[\/latex]. In the following exercises (9-10), for each value of [latex]\\varepsilon[\/latex], find a value of [latex]\\delta >0[\/latex] such that the precise definition of limit holds true.<\/p>\n

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

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9.\u00a0<\/strong>[latex]\\varepsilon =1.5[\/latex]<\/p>\n<\/div>\n<\/div>\n

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10.\u00a0<\/strong>[latex]\\varepsilon =3[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\delta \\le 1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises (11-12), use a graphing calculator to find a number [latex]\\delta[\/latex] such that the statements hold true.<\/p>\n

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11. [T]\u00a0<\/strong>[latex]|\\sin (2x)-\\frac{1}{2}|<0.1[\/latex], whenever [latex]|x-\\frac{\\pi}{12}|<\\delta[\/latex]<\/p>\n<\/div>\n<\/div>\n

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12. [T]\u00a0<\/strong>[latex]|\\sqrt{x-4}-2|<0.1[\/latex], whenever [latex]|x-8|<\\delta[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\delta <0.3900[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises (13-17), use the precise definition of limit to prove the given limits.<\/p>\n

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13.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}(5x+8)=18[\/latex]<\/p>\n<\/div>\n<\/div>\n

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14.\u00a0<\/strong>[latex]\\underset{x\\to 3}{\\lim}\\dfrac{x^2-9}{x-3}=6[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Let [latex]\\delta =\\varepsilon[\/latex]. If [latex]0<|x-3|<\\varepsilon[\/latex], then [latex]|x+3-6|=|x-3|<\\varepsilon[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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15.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}\\dfrac{2x^2-3x-2}{x-2}=5[\/latex]<\/p>\n<\/div>\n<\/div>\n

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16.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x^4=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Let [latex]\\delta =\\sqrt[4]{\\varepsilon}[\/latex]. If [latex]0<|x|<\\sqrt[4]{\\varepsilon}[\/latex], then [latex]|x^4|=x^4<\\varepsilon[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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17.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}(x^2+2x)=8[\/latex]<\/p>\n<\/div>\n<\/div>\n

In the following exercises (18-20), use the precise definition of limit to prove the given one-sided limits.<\/p>\n

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18.\u00a0<\/strong>[latex]\\underset{x\\to 5^-}{\\lim}\\sqrt{5-x}=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Let [latex]\\delta =\\varepsilon^2[\/latex]. If [latex]5-\\varepsilon^2\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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19.\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}f(x)=-2[\/latex], where [latex]f(x)=\\begin{cases} 8x-3 & \\text{ if } \\, x<0 \\\\ 4x-2 & \\text{ if } \\, x \\ge 0 \\end{cases}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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20.\u00a0<\/strong>[latex]\\underset{x\\to 1^-}{\\lim}f(x)=3[\/latex], where [latex]f(x)=\\begin{cases} 5x-2 & \\text{ if } \\, x < 1 \\\\ 7x-1 & \\text{ if } x \\ge 1 \\end{cases}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Let [latex]\\delta =\\varepsilon\/5[\/latex]. If [latex]1-\\varepsilon\/5\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises (21-23), use the precise definition of limit to prove the given infinite limits.<\/p>\n

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21.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\dfrac{1}{x^2}=\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n

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22.\u00a0<\/strong>[latex]\\underset{x\\to -1}{\\lim}\\dfrac{3}{(x+1)^2}=\\infty[\/latex]<\/p>\n

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Let [latex]\\delta =\\sqrt{\\frac{3}{N}}[\/latex]. If [latex]0<|x+1|<\\sqrt{\\frac{3}{N}}[\/latex], then [latex]f(x)=\\frac{3}{(x+1)^2}>N[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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23.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}-\\dfrac{1}{(x-2)^2}=\u2212\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n

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24.\u00a0<\/strong>An engineer is using a machine to cut a flat square of Aerogel of area 144 cm2<\/sup>. If there is a maximum error tolerance in the area of 8 cm2<\/sup>, how accurately must the engineer cut on the side, assuming all sides have the same length? How do these numbers relate to [latex]\\delta, \\, \\varepsilon, \\, a[\/latex], and [latex]L[\/latex]?<\/p>\n

Show Solution<\/span><\/p>\n
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0.033 cm, [latex]\\varepsilon =8, \\, \\delta =0.33, \\, a=12, \\, L=144[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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25.\u00a0<\/strong>Use the precise definition of limit to prove that the following limit does not exist: [latex]\\underset{x\\to 1}{\\lim}\\dfrac{|x-1|}{x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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26.\u00a0<\/strong>Using precise definitions of limits, prove that [latex]\\underset{x\\to 0}{\\lim}f(x)[\/latex] does not exist, given that [latex]f(x)[\/latex] is the ceiling function.<\/p>\n

Hint<\/span><\/p>\n
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Try any [latex]\\delta <1[\/latex].\n\n<\/div>\n<\/div>\n

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Answers may vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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27.\u00a0<\/strong>Using precise definitions of limits, prove that [latex]\\underset{x\\to 0}{\\lim}f(x)[\/latex] does not exist: [latex]f(x)=\\begin{cases} 1 & \\text{ if } \\, x \\, \\text{is rational} \\\\ 0 & \\text{ if } \\, x \\, \\text{is irrational} \\end{cases}[\/latex]<\/p>\n

Hint<\/span><\/p>\n
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Think about how you can always choose a rational number [latex]0\n<\/div>\n<\/div>\n<\/div>\n

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28.\u00a0<\/strong>Using precise definitions of limits, determine [latex]\\underset{x\\to 0}{\\lim}f(x)[\/latex] for [latex]f(x)=\\begin{cases} x & \\text{ if } \\, x \\, \\text{is rational} \\\\ 0 & \\text{ if } \\, x \\, \\text{is irrational} \\end{cases}[\/latex]<\/p>\n

Hint<\/span><\/p>\n
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Break into two cases, [latex]x[\/latex] rational and [latex]x[\/latex] irrational.<\/p>\n<\/div>\n<\/div>\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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29.\u00a0<\/strong>Using the function from the previous exercise, use the precise definition of limits to show that [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] does not exist for [latex]a\\ne 0[\/latex].<\/p>\n<\/div>\n<\/div>\n

For the following exercises (30-32), suppose that [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=M[\/latex] both exist. Use the precise definition of limits to prove the following limit laws:<\/p>\n

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30.\u00a0<\/strong>[latex]\\underset{x\\to a}{\\lim}(f(x)-g(x))=L-M[\/latex]<\/p>\n

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Show Solution<\/span><\/p>\n
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[latex]f(x)-g(x)=f(x)+(-1)g(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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31.\u00a0<\/strong>[latex]\\underset{x\\to a}{\\lim}[cf(x)]=cL[\/latex] for any real constant [latex]c[\/latex]<\/p>\n

Hint<\/span><\/p>\n
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Consider two cases: [latex]c=0[\/latex] and [latex]c\\ne 0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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32.\u00a0<\/strong>[latex]\\underset{x\\to a}{\\lim}[f(x)g(x)]=LM[\/latex].<\/p>\n

Hint<\/span><\/p>\n
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[latex]|f(x)g(x)-LM|=|f(x)g(x)-f(x)M+f(x)M-LM|\\le |f(x)||g(x)-M|+|M||f(x)-L|[\/latex].<\/p>\n<\/div>\n<\/div>\n

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