{"id":490,"date":"2021-02-04T15:31:12","date_gmt":"2021-02-04T15:31:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=490"},"modified":"2021-06-24T04:33:25","modified_gmt":"2021-06-24T04:33:25","slug":"problem-set-lhopitals-rule","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-lhopitals-rule\/","title":{"raw":"Problem Set: L\u2019H\u00f4pital\u2019s Rule","rendered":"Problem Set: L\u2019H\u00f4pital\u2019s Rule"},"content":{"raw":"

For the following exercises (1-6), evaluate the limit.<\/p>\r\n\r\n

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1.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x^k}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1165043218009\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043218009\"]\r\n

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

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3.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{x^k}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^2-a^2}, \\,\\ a\\ne 0[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1165043218117\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043218117\"]\r\n

[latex]\\frac{1}{2a}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^3-a^3}, \\,\\ a\\ne 0[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^n-a^n}, \\,\\ a\\ne 0[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1165042469706\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042469706\"]\r\n

[latex]\\frac{1}{na^{n-1}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (7-11), determine whether you can apply L\u2019H\u00f4pital\u2019s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L\u2019H\u00f4pital\u2019s rule.<\/p>\r\n\r\n

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7.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim}x^2 \\ln x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n8.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x^{1\/x}[\/latex]\r\n\r\n[reveal-answer q=\"fs-id1165042711556\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711556\"]\r\n

Cannot apply directly; use logarithms<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.<\/strong> [latex]\\underset{x\\to 0}{\\lim} x^{2\/x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{x^2}{1\/x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042711643\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711643\"]\r\n

Cannot apply directly; rewrite as [latex]\\underset{x\\to 0}{\\lim} x^3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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11.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (12-40), evaluate the limits with either L\u2019H\u00f4pital\u2019s rule or previously learned methods.<\/p>\r\n\r\n

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12.<\/strong> [latex]\\underset{x\\to 3}{\\lim}\\frac{x^2-9}{x-3}[\/latex]<\/p>\r\n\r\n

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

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13.<\/strong> [latex]\\underset{x\\to 3}{\\lim}\\frac{x^2-9}{x+3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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14.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^{-2}-1}{x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042499552\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042499552\"]\r\n

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

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15.<\/strong> [latex]\\underset{x\\to \\pi \/2}{\\lim}\\frac{\\cos x}{\\pi \/ 2 - x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.<\/strong> [latex]\\underset{x\\to \\pi }{\\lim}\\frac{x-\\pi }{\\sin x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042499655\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042499655\"]\r\n

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

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17.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{\\sin x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^n-1}{x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042711303\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711303\"]\r\n

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

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\r\n\r\n19.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^n-1-nx}{x^2}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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20.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x- \\tan x}{x^3}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042711441\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711441\"]\r\n

[latex]-\\frac{1}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sqrt{1+x}-\\sqrt{1-x}}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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22.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-x-1}{x^2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042602950\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042602950\"]\r\n

[latex]\\frac{1}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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23.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\tan x}{\\sqrt{x}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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24.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{\\ln x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042603044\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042603044\"]\r\n

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

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25.<\/strong> [latex]\\underset{x\\to 0}{\\lim}(x+1)^{1\/x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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26.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{\\sqrt{x}-\\sqrt[3]{x}}{x-1}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042617540\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042617540\"]\r\n

[latex]\\frac{1}{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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27.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim} x^{2x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x \\sin (\\frac{1}{x})[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042617638\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042617638\"]\r\n

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

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29.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x-x}{x^2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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30.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim} x \\ln (x^4)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042617740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042617740\"]\r\n

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

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

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32.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x^2 e^{\u2212x}[\/latex]<\/p>\r\n[reveal-answer q=\"849216\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"849216\"]0[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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33.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{3^x-2^x}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n34.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{1+1\/x}{1-1\/x}[\/latex]\r\n\r\n[reveal-answer q=\"fs-id1165042707893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042707893\"]\r\n

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

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35.<\/strong> [latex]\\underset{x\\to \\pi \/4}{\\lim}(1- \\tan x) \\cot x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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36.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} xe^{1\/x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042525331\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042525331\"]\r\n

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

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37.<\/strong> [latex]\\underset{x\\to 0}{\\lim} x^{1\/ \\cos x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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38.<\/strong> [latex]\\underset{x\\to 0^{+}}{\\lim} x^{1\/x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042525418\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042525418\"]0[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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39.<\/strong> [latex]\\underset{x\\to 0}{\\lim} (1-\\frac{1}{x})^x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} (1-\\frac{1}{x})^x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042525526\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042525526\"]\r\n

[latex]\\frac{1}{e}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (41-50), use a calculator to graph the function and estimate the value of the limit, then use L\u2019H\u00f4pital\u2019s rule to find the limit directly.<\/p>\r\n\r\n

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\r\n\r\n41. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-1}{x}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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42. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x \\sin (\\frac{1}{x})[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043382979\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043382979\"]\r\n

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

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43. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{1- \\cos (\\pi x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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44. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{e^{x-1}-1}{x-1}[\/latex]<\/p>\r\n[reveal-answer q=\"855100\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"855100\"]1[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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45. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{(x-1)^2}{\\ln x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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46. [T]\u00a0<\/strong>[latex]\\underset{x\\to \\pi }{\\lim}\\frac{1+ \\cos x}{ \\sin x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042710220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042710220\"]\r\n

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

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47. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}( \\csc x-\\frac{1}{x})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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48. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim} \\tan (x^x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042710332\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042710332\"]\r\n

[latex] \\tan (1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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49. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{ \\sin x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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50. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-e^{\u2212x}}{x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042539145\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042539145\"]\r\n

2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

For the following exercises (1-6), evaluate the limit.<\/p>\n

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1.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n

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2.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x^k}[\/latex].<\/p>\n

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

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3.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{x^k}[\/latex].<\/p>\n<\/div>\n<\/div>\n

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4.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^2-a^2}, \\,\\ a\\ne 0[\/latex].<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{1}{2a}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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5.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^3-a^3}, \\,\\ a\\ne 0[\/latex].<\/p>\n<\/div>\n<\/div>\n

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6.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^n-a^n}, \\,\\ a\\ne 0[\/latex].<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{1}{na^{n-1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (7-11), determine whether you can apply L\u2019H\u00f4pital\u2019s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L\u2019H\u00f4pital\u2019s rule.<\/p>\n

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7.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim}x^2 \\ln x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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8.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x^{1\/x}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Cannot apply directly; use logarithms<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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9.<\/strong> [latex]\\underset{x\\to 0}{\\lim} x^{2\/x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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10.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{x^2}{1\/x}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Cannot apply directly; rewrite as [latex]\\underset{x\\to 0}{\\lim} x^3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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11.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (12-40), evaluate the limits with either L\u2019H\u00f4pital\u2019s rule or previously learned methods.<\/p>\n

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12.<\/strong> [latex]\\underset{x\\to 3}{\\lim}\\frac{x^2-9}{x-3}[\/latex]<\/p>\n

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

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13.<\/strong> [latex]\\underset{x\\to 3}{\\lim}\\frac{x^2-9}{x+3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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14.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^{-2}-1}{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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15.<\/strong> [latex]\\underset{x\\to \\pi \/2}{\\lim}\\frac{\\cos x}{\\pi \/ 2 - x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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16.<\/strong> [latex]\\underset{x\\to \\pi }{\\lim}\\frac{x-\\pi }{\\sin 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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17.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{\\sin x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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18.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^n-1}{x}[\/latex]<\/p>\n

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

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19.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^n-1-nx}{x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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20.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x- \\tan x}{x^3}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]-\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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21.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sqrt{1+x}-\\sqrt{1-x}}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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22.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-x-1}{x^2}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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23.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\tan x}{\\sqrt{x}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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24.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{\\ln x}[\/latex]<\/p>\n

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1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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25.<\/strong> [latex]\\underset{x\\to 0}{\\lim}(x+1)^{1\/x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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26.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{\\sqrt{x}-\\sqrt[3]{x}}{x-1}[\/latex]<\/p>\n

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[latex]\\frac{1}{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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27.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim} x^{2x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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28.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x \\sin (\\frac{1}{x})[\/latex]<\/p>\n

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1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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29.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x-x}{x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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30.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim} x \\ln (x^4)[\/latex]<\/p>\n

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0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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31.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}(x-e^x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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32.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x^2 e^{\u2212x}[\/latex]<\/p>\n

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0<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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33.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{3^x-2^x}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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34.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{1+1\/x}{1-1\/x}[\/latex]<\/p>\n

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-1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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35.<\/strong> [latex]\\underset{x\\to \\pi \/4}{\\lim}(1- \\tan x) \\cot x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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36.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} xe^{1\/x}[\/latex]<\/p>\n

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[latex]\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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37.<\/strong> [latex]\\underset{x\\to 0}{\\lim} x^{1\/ \\cos x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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38.<\/strong> [latex]\\underset{x\\to 0^{+}}{\\lim} x^{1\/x}[\/latex]<\/p>\n

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0<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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39.<\/strong> [latex]\\underset{x\\to 0}{\\lim} (1-\\frac{1}{x})^x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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40.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} (1-\\frac{1}{x})^x[\/latex]<\/p>\n

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[latex]\\frac{1}{e}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (41-50), use a calculator to graph the function and estimate the value of the limit, then use L\u2019H\u00f4pital\u2019s rule to find the limit directly.<\/p>\n

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41. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-1}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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42. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x \\sin (\\frac{1}{x})[\/latex]<\/p>\n

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0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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43. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{1- \\cos (\\pi x)}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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44. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{e^{x-1}-1}{x-1}[\/latex]<\/p>\n

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1<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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45. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{(x-1)^2}{\\ln x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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46. [T]\u00a0<\/strong>[latex]\\underset{x\\to \\pi }{\\lim}\\frac{1+ \\cos x}{ \\sin x}[\/latex]<\/p>\n

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0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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47. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}( \\csc x-\\frac{1}{x})[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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[latex]\\tan (1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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49. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{ \\sin x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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50. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-e^{\u2212x}}{x}[\/latex]<\/p>\n

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2<\/p>\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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