Answer: Calculus History In LHpitals 1696 calculus textbook, he illustrated his rule

Numerical and Graphical Analysis In Exercises 1-4, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 3 x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0 sin 4x / sin 3x

Numerical and Graphical Analysis In Exercises 1-4, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. \(\lim _{x \rightarrow 0} \frac{1-e^{x}}{x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0 sin 1 - e^x / x

Numerical and Graphical Analysis In Exercises 1-4, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. \(\lim _{x \rightarrow \infty} x^{5} e^{-x / 100}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity x^5 e^-x/100

Numerical and Graphical Analysis In Exercises 1-4, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. \(\lim _{x \rightarrow \infty} \frac{6 x}{\sqrt{3 x^{2}-2 x}}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity 6x / sqrt 3x^2 - 2x

Using Two Methods In Exercises 5–10,evaluate the limit (a) using techniques from Chapters 2 and 4 and (b) using L’Hôpital’s Rule. \(\lim _{x \rightarrow 4} \frac{3(x-4)}{x^{2}-16}\) Equation Transcription: Text Transcription: lim_x rightarrow 4 3(x-4) / x^2-16

Using Two Methods In Exercises 5–10,evaluate the limit (a) using techniques from Chapters 2 and 4 and (b) using L’Hôpital’s Rule. \(\lim _{x \rightarrow-4} \frac{2 x^{2}+13 x+20}{x+4}\) Equation Transcription: Text Transcription: lim_x rightarrow -4 2x^2+13x+20/x+4

Using Two Methods In Exercises 5–10,evaluate the limit (a) using techniques from Chapters 2 and 4 and (b) using L’Hôpital’s Rule. \(\lim _{x \rightarrow 6} \frac{\sqrt{x+10}-4}{x-6}\) Equation Transcription: Text Transcription: lim_x rightarrow 6 sqrt x+10 - 4/x-6

Using Two Methods In Exercises 5–10,evaluate the limit (a) using techniques from Chapters 2 and 4 and (b) using L’Hôpital’s Rule. \(\lim _{x \rightarrow 0} \frac{\sin 6 x}{4 x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0 sin 6x/4x

Using Two Methods In Exercises 5–10,evaluate the limit (a) using techniques from Chapters 2 and 4 and (b) using L’Hôpital’s Rule. \(\lim _{x \rightarrow \infty} \frac{5 x^{2}-3 x+1}{3 x^{2}-5}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity 5x^2-3x+1/3x^2-5

Using Two Methods In Exercises 5–10,evaluate the limit (a) using techniques from Chapters 2 and 4 and (b) using L’Hôpital’s Rule. \(\lim _{x \rightarrow \infty} \frac{4 x-3}{5 x^{2}+1}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity 4x-3/5x^2+1

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 3} \frac{x^{2}-2 x-3}{x-3}\) Equation Transcription: Text Transcription: lim_x rightarrow 3 x^2-2x-3/x-3

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow-2} \frac{x^{2}-3 x-10}{x+2}\) Equation Transcription: Text Transcription: lim_x rightarrow -2 x^2-3x-10/x+2

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 0} \frac{\sqrt{25-x^{2}}-5}{x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0 sqrt 25-x^2 - 5/x

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 5^{-}} \frac{\sqrt{25-x^{2}}}{x-5}\) Equation Transcription: Text Transcription: lim_x rightarrow 5^- sqrt 25-x^2/x-5

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 0^{+}} \frac{e^{x}-(1+x)}{x^{3}}\) Equation Transcription: Text Transcription: lim_x rightarrow 0^+ e^x-(1+x)/x^3

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 1} \frac{\ln x^{3}}{x^{2}-1}\) Equation Transcription: Text Transcription: lim_x rightarrow 1 ln x^3/x^2-1

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 1} \frac{x^{11}-1}{x^{4}-1}\) Equation Transcription: Text Transcription: lim_x rightarrow 1 x^11 - 1/x^4-1

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 1} \frac{x^{a}-1}{x^{b}-1}, \text { where } a, b \neq 0\) Equation Transcription: Text Transcription: lim_x rightarrow 1 x^a-1/x^b-a, where a,b neq 0

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 0} \frac{\sin 3 x}{\sin 5 x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0 sin 3x/sin 5x

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}, \text { where } a, b \neq 0\) Equation Transcription: Text Transcription: lim_x rightarrow 0 sin ax/sin bx, where a,b neq 0

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 0} \frac{\arcsin x}{x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0 arcsin x/x

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 1} \frac{\arcsin x-(\pi / 4)}{x-1}\) Equation Transcription: Text Transcription: lim_x rightarrow 1 arcsin x-(pi/4)/x-1

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{5 x^{2}+3 x-1}{4 x^{2}+5}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity 5x^2+3x-1/4x^2+5

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{5 x+3}{x^{3}-6 x+2}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity 5x+3/x^3-6x+2

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{x^{2}+4 x+7}{x-6}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity x^2+4x+7/x-6

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{x^{3}}{x+2}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity x^3/x+2

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{x / 2}}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity x^3/e^x/2

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{x^{2}}}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity x^3/e^x^2

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity x/sqrt x^2+1

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{x^{2}}{\sqrt{x^{2}+1}}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity x^2/sqrt x^2+1

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{\cos x}{x}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity cos x/x

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{\sin x}{x-\pi}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity sin x/x-pi

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{\ln x}{x^{2}}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity ln x/x^2

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{\ln x^{4}}{x^{3}}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity ln x^4/x^3

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{e^{x}}{x^{4}}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity e^x/x^4

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{e^{x / 2}}{x}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity e^x/2/x

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 0} \frac{\sin 5 x}{\tan 9 x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0 sin 5x/tan 9x

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 1} \frac{\ln x}{\sin \pi x}\) Equation Transcription: Text Transcription: lim_x rightarrow 1 ln x/sin pi x

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 0} \frac{\arctan x}{\sin x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0 arctan x/sin x

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 0} \frac{x}{\arctan 2 x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0 x/arctan 2x

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow \infty} \frac{\int_{1}^{x} \ln \left(e^{4-1}\right) d t}{x}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity int^x_1 ln(e^4-1)dt/x

Evaluating a Limit In Exercises 11–42, evaluate the limit, using L’Hôpital’s Rule if necessary. \(\lim _{x \rightarrow 1^{+}} \frac{\int_{1}^{x} \cos \theta \ d \theta}{x-1}\) Equation Transcription: Text Transcription: lim_x rightarrow 1^+ int^x_1 cos theta d theta/x-1

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow \infty} x \ln x\) Equation Transcription: Text Transcription: lim_x rightarrow infinity x ln x

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 0^{+}} x^{3} \cot x\) Equation Transcription: Text Transcription: lim_x rightarrow 0^+ x^3 cot x

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow \infty}\left(x \sin \frac{1}{x}\right)\) Equation Transcription: Text Transcription: lim_x rightarrow infinity (x sin 1/x)

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow \infty} x \tan \frac{1}{x}\) Equation Transcription: Text Transcription: lim_x rightarrow infinity x tan 1/x

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 0^{+}} x^{1 / x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0^+ x^1/x

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 0^{+}}\left(e^{x}+x\right)^{2 / x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0^+ (e^x+x)^2/x

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow \infty} x^{1 / x}\) Text Transcription: lim_x rightarrow infinity x^1/x

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}\) Text Transcription: lim_x rightarrow infinity (1+1/x)^x

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 0^{+}}(1+x)^{1 / x}\) Text Transcription: lim_x rightarrow 0^+ (1 + x)^1/x

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow \infty}(1+x)^{1 / x}\) Text Transcription: lim_x rightarrow infinity (1 + x)^1/x

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 0^{+}}\left[3(x)^{x / 2}\right]\) Text Transcription: lim_x rightarrow 0^+ [3(x)^x/2]

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 4^{+}}[3(x-4)]^{x-4}\) Text Transcription: lim_x rightarrow 4^+ [3(x-4)]^x-4

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 1^{+}}(\ln x)^{x-1}\) Text Transcription: lim_x rightarrow 1^+ (ln x)^x-1

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 0^{+}}\left[\cos \left(\frac{\pi}{2}-x\right)\right]^{x}\) Text Transcription: lim_x rightarrow 0^+ [cos(pi/2 - x)]^x

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 2^{+}}\left(\frac{8}{x^{2}-4}-\frac{x}{x-2}\right)\) Text Transcription: lim_x rightarrow 2^+ (8/x^2-4 - x/x-2)

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 2^{+}}\left(\frac{1}{x^{2}-4}-\frac{\sqrt{x-1}}{x^{2}-4}\right)\) Text Transcription: lim_x rightarrow 2^+ (1/x^2-4 - sqrt x - 1/x^2-4)

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 1^{+}}\left(\frac{3}{\ln x}-\frac{2}{x-1}\right)\) Text Transcription: lim_x rightarrow 1^+ (3/ln x - 2/x-1)

Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 0^{+}}\left(\frac{10}{x}-\frac{3}{x^{2}}\right)\) Text Transcription: lim_x rightarrow 0^+ (10/x - 3/x^2)

Indeterminate Forms List six different indeterminate forms.

LHpitals Rule State LHpitals Rule.

Finding Functions Find differentiable functions f and g that satisfy the specified condition such that \(\lim _{x \rightarrow 5} f(x)=0 \quad \text { and } \quad \lim _{x \rightarrow 5} g(x)=0 .\) Explain how you obtained your answers. (Note: There are many correct answers.) (a) \(\lim _{x \rightarrow 5} \frac{f(x)}{g(x)}=10\) (b) \(\lim _{x \rightarrow 5} \frac{f(x)}{g(x)}=0\) (c) \(\lim _{x \rightarrow 5} \frac{f(x)}{g(x)}=\infty\) Text Transcription: lim_x rightarrow 5 f(x)=0 and lim_rightarrow 5 g(x)=0 lim_x rightarrow 5 f(x)/g(x)=10 lim_x rightarrow 5 f(x)/g(x)=0 lim_x rightarrow 5 f(x)/g(x)=infinity

Finding Functions Find differentiable functions f and g such that \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow \infty} g(x)=\infty \text { and } \lim _{x \rightarrow \infty}[f(x)-g(x)]=25\). Explain how you obtained your answers. (Note: There are many correct answers.) Text Transcription: lim_x rightarrow infinity f(x)=lim_x rightarrow infinity g(x)=infinity and lim_x rightarrow infinity [f(x)-g(x)]=25

L’Hôpital’s Rule Determine which of the following limits can be evaluated using L’Hôpital’s Rule. Explain your reasoning. Do not evaluate the limit. (a) \(\lim _{x \rightarrow 2} \frac{x-2}{x^{3}-x-6}\) (b) \(\lim _{x \rightarrow 0} \frac{x^{2}-4 x}{2 x-1}\) (c) \(\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{x}}\) (d) \(\lim _{x \rightarrow 3} \frac{e^{x^{2}}-e^{9}}{x-3}\) (e) \(\lim _{x \rightarrow 1} \frac{\cos \pi x}{\ln x}\) (f) \(\lim _{x \rightarrow 1} \frac{1+x(\ln x-1)}{(x-1) \ln x}\) Text Transcription: lim_x rightarrow 2 x-2/x^3-x-6 lim_x rightarrow 0 x^2-4x/2x-1 lim_x rightarrow infinity x^3/e^x lim_x rightarrow 3 e^x^2-e^9/x-3 lim_x rightarrow 1 cos pi x/ln x lim_x rightarrow 1 1+x(ln x-1)/(x-1)ln x

HOW DO YOU SEE IT? Use the graph of f to find the limit. (a) \(\lim _{x \rightarrow 1^{-}} f(x)\) (b) \(\lim _{x \rightarrow 1^{+}} f(x)\) (c) \(\lim _{x \rightarrow 1} f(x)\) Text Transcription: lim_x rightarrow 1^- f(x) lim_x rightarrow 1^+ f(x) lim_x rightarrow 1 f(x)

Numerical Approach Complete the table to show that eventually “overpowers” \((ln x)^x\). Text Transcription: (ln x)^x

Numerical Approach Complete the table to show that eventually “overpowers” \(x^5\). Text Transcription: x^5

Comparing Functions In Exercises 69-74, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}, \quad g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, and \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{x^{2}}{e^{5 x}}\) Text Transcription: f(x)=x^m, g(x)=e^nx h(x)=(ln x)^n x rightarrow infinity lim_x rightarrow infinity x^2/e^5x

Comparing Functions In Exercises 69-74, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}, \quad g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, and \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{2 x}}\) Text Transcription: f(x)=x^m, g(x)=e^nx h(x)=(ln x)^n x rightarrow infinity lim_x rightarrow infinity x^3/e^2x

Comparing Functions In Exercises 69-74, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}, \quad g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, and \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{(\ln x)^{3}}{x}\) Text Transcription: f(x)=x^m, g(x)=e^nx h(x)=(ln x)^n x rightarrow infinity lim_x rightarrow infinity (ln x)^3/x

Comparing Functions In Exercises 69-74, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}, \quad g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, and \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{(\ln x)^{2}}{x^{3}}\) Text Transcription: f(x)=x^m, g(x)=e^nx h(x)=(ln x)^n x rightarrow infinity lim_x rightarrow infinity (ln x)^2/x^3

Comparing Functions In Exercises 69-74, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}, \quad g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, and \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{(\ln x)^{n}}{x^{m}}\) Text Transcription: f(x)=x^m, g(x)=e^nx h(x)=(ln x)^n x rightarrow infinity lim_x rightarrow infinity (ln x)^n/x^m

Comparing Functions In Exercises 69-74, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}, \quad g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, and \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{x^{m}}{e^{n x}}\) Text Transcription: f(x)=x^m, g(x)=e^nx h(x)=(ln x)^n x rightarrow infinity lim_x rightarrow infinity x^m/e^nx

Asymptotes and Relative Extrema In Exercises 75-78, find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in previous exercises.) \(y=x^{1 / x}, \quad x>0\) Text Transcription: y=x^1/x, x > 0

Asymptotes and Relative Extrema In Exercises 75-78, find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in previous exercises.) \(y=x^{x}, \quad x>0\) Text Transcription: y=x^x, x > 0

Asymptotes and Relative Extrema In Exercises 75-78, find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in previous exercises.) \(y=2 x e^{-x}\) Text Transcription: y=2xe^-x

Asymptotes and Relative Extrema In Exercises 75-78, find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in previous exercises.) \(y=\frac{\ln x}{x}\) Text Transcription: y=ln x/x

Think About It In Exercises 79-82, L’Hôpital’s Rule is used incorrectly. Describe the error. \(\lim _{x \rightarrow 2} \frac{3 x^{2}+4 x+1}{x^{2}-x-2}=\lim _{x \rightarrow 2} \frac{6 x+4}{2 x-1}=\lim _{x \rightarrow 2} \frac{6}{2}=3\) Text Transcription: lim_x rightarrow 2 3x^+4x+1/x^2-x-2 = lim_x righarrow 1 6x+4/2x-1 = lim_x rightarrow 2 6/2 = 3

Think About It In Exercises 79-82, L’Hôpital’s Rule is used incorrectly. Describe the error. \(\begin{array}{l} \lim _{x \rightarrow 0} \frac{e^{2 x}-1}{e^{x}}=\lim _{x 0} \frac{2 e^{2 x}}{e^{x}} \\ =\lim _{x \rightarrow 0} 2 e^{x} \\ =2 \end{array} \) Text Transcription: lim_x rightarrow 0 e^2x-1/e^x = lim_x rightarrow 0 2e^2x/e^x =lim_x rightarrow 0 2e^x =2

Think About It In Exercises 79-82, L’Hôpital’s Rule is used incorrectly. Describe the error. \(\begin{array}{l} \lim _{x \rightarrow \infty} \frac{e^{-x}}{1+e^{-x}}=\lim _{x \rightarrow \infty} \frac{-e^{-x}}{-e^{-x}} \\ =\lim _{x \rightarrow \infty} 1 \\ =1 \end{array}\) Text Transcription: lim_x rightarrow infinity e^-x/1+e^-x=lim_x rightarrow infinity -e^-x/-e^-x =lim_x rightarrow infinity 1 =1

Think About It In Exercises 79-82, L’Hôpital’s Rule is used incorrectly. Describe the error. \(\begin{array}{l} \lim _{x \rightarrow \infty} \operatorname{xcos} \frac{1}{x}=\lim _{x \rightarrow \infty} \frac{\cos (1 / x)}{1 / x} \\ =\lim _{x \rightarrow \infty} \frac{-\sin (1 / x)]\left(1 / x^{2}\right)}{1 / x^{2}} \\ =0 \end{array}\) Text Transcription: lim_x rightarrow infinity x cos 1/x=lim_x rightarrow infinity cos(1/x)/1/x =lim_x rightarrow infinity [-sin(1/x)](1.x^2)/-1/x^2 =0

Analytical Approach In Exercises 83 and 84, (a) explain why L’Hôpital’s Rule cannot be used to find the limit, (b) find the limit analytically, and (c) use a graphing utility to graph the function and approximate the limit from the graph. Compare the result with that in part (b). \(\lim _{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}}\) Text Transcription: lim_x rightarrow infinity x/sqrt x^2+1

Analytical Approach In Exercises 83 and 84, (a) explain why L’Hôpital’s Rule cannot be used to find the limit, (b) find the limit analytically, and (c) use a graphing utility to graph the function and approximate the limit from the graph. Compare the result with that in part (b). \(\lim _{x \rightarrow \pi / 2^{-}} \frac{\tan x}{\sec x}\) Text Transcription: lim_x rightarrow pi/2^- tan x/sec x

Graphical Analysis In Exercises 85 and 86, graph f(x)/g(x) and f’(x)/g’(x) near x = 0. What do you notice about these ratios as \(x \rightarrow 0\)? How does this illustrate L’Hôpital’s Rule? f(x) = sin 3x, g(x = sin 4x Text Transcription: x rightarrow 0

Graphical Analysis In Exercises 85 and 86, graph f(x)/g(x) and f’(x)/g’(x) near x = 0. What do you notice about these ratios as \(x \rightarrow 0\)? How does this illustrate L’Hôpital’s Rule? \(f(x)=e^{3 x}-1, \quad g(x)=x\) Text Transcription: x rightarrow 0 f(x)=e^3x -1, g(x) = x

Velocity in a Resisting Medium The velocity of an object falling through a resisting medium such as air or water is given by \(v=\frac{32}{k}\left(1-e^{-k t}+\frac{v_{0} k e^{-k t}}{32}\right)\) where \(v_0\) is the initial velocity, is the time in seconds, and k is the resistance constant of the medium. Use L’Hôpital’s Rule to find the formula for the velocity of a falling body in a vacuum by fixing \(v_0\) and t and letting k approach zero. (Assume that the downward direction is positive.) Text Transcription: v=32/k(1-e^-kt+v_0 ke^-kt/32) v_0

Compound Interest The formula for the amount A n a savings account compounded n times per year for t years at an interest rate r and an initial deposit of P is given by \(A=P\left(1+\frac{r}{n}\right)^{n t}\) Use L’Hôpital’s Rule to show that the limiting formula as the number of compoundings per year approaches infinity is given by \(A=P e^{r t}\) Text Transcription: A=P(1+r/n)^nt A=Pe^rt

The Gamma function The Gamma Functions \(\Gamma(n)\) is defined in terms of the integral of the function given by \(f(x)=x^{n-1} e^{-x}, \quad n>0\). Show that for any fixed value of n, the limit of f(x) as x approaches infinity is zero. Text Transcription: gamma(n) f(x)=x^n-1 e^-x, n>0

Tractrix A person moves from the origin along the positive y-axis pulling a weight at the end of a 12-meter rope (see figure). Initially, the weight is located at the point (12,0). (a) Show that the slope of the tangent line of the path of the weight is \(\frac{d y}{d x}=-\frac{\sqrt{144-x^{2}}}{x}\) (b) Use the result of part (a) to find the equation of the path of the weight. Use a graphing utility to graph the path and compare it with the figure. (c) Find any vertical asymptotes of the graph in part (b). (d) When the person has reached the point (0,12), how far has the weight moved? Text Transcription: dy/dx = -sqrt 144-x^2/x

Extended Mean Value Theorem In Exercises 91-94, apply the Extended Mean Value Theorem to the functions f and g on the given interval. Find all values in the interval (a,b) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\) Function Interval \(f(x)=x^{3}, \quad g(x)=x^{2}+1\) [0, 1] Text Transcription: f’(c)/g’(c)=f(b)-f(a)/g(b)-g(a) f(x)=x^3, g(x)=x^2+1

Extended Mean Value Theorem In Exercises 91-94, apply the Extended Mean Value Theorem to the functions f and g on the given interval. Find all values in the interval (a,b) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\) Function Interval \(f(x)=\frac{1}{x}, \quad g(x)=x^{2}-4\) [1, 2] Text Transcription: f’(c)/g’(c)=f(b)-f(a)/g(b)-g(a) f(x)=1/x, g(x)=x^2-4

Extended Mean Value Theorem In Exercises 91-94, apply the Extended Mean Value Theorem to the functions f and g on the given interval. Find all values in the interval (a,b) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\) Function Interval f(x) = sin x, g(x) = cos x \(\left[0, \frac{\pi}{2}\right]\) Text Transcription: f’(c)/g’(c)=f(b)-f(a)/g(b)-g(a) [0, pi/2]

Extended Mean Value Theorem In Exercises 91-94, apply the Extended Mean Value Theorem to the functions f and g on the given interval. Find all values in the interval (a,b) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\) Function Interval \(f(x)=\ln x, \quad g(x)=x^{3}\) [1, 4] Text Transcription: f’(c)/g’(c)=f(b)-f(a)/g(b)-g(a) f(x)=ln x, g(x)=x^3

True or False? In Exercises 95-98, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(\lim _{x \rightarrow 0}\left[\frac{x^{2}+x+1}{x}\right]=\lim _{x \rightarrow 0}\left[\frac{2 x+1}{1}\right]=1\) Text Transcription: lim_x rightarrow 0[x^2+x+1/x]=lim_x rightarrow 0[2x+1/1]=1

True or False? In Exercises 95-98, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(\text { If } y=\frac{e^{x}}{x^{2}} \text {, then } y^{\prime}=\frac{e^{x}}{2 x} \text {. }\) Text Transcription: If y=e^x/x^2, then y’=e^x/2x

True or False? In Exercises 95-98, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If p(x) is a polynomial, then \(\lim _{x \rightarrow \infty} \frac{p(x)}{e^{x}}=0\). Text Transcription: lim_x rightarrow infinity p(x)/e^x=0

True or False? In Exercises 95-98, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(\text { If } \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=1 \text {, then } \lim _{x \rightarrow \infty}[f(x)-g(x)]=0 \text {. }\) Text Transcription: If lim_x rightarrow infinity f(x)/g(x)=1, then lim_x rightarrow infinity [f(x)-g(x)]=0

Area Find the limit, as x approaches 0, of the ratio of the area of the triangle to the total shaded area in the figure.

Finding a Limit In Section 2.3, a geometric argument (see figure) was used to prove that \(\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1\) (a) Write the area of \(\triangle A B D\) terms of \(\theta\). (b) Write the area of the shaded region in terms of \(\theta\). (c) Write the ratio R of the area \(\triangle A B D\) to that of the shaded region. (d) Find \(\lim _{\theta \rightarrow 0} R\) Text Transcription: triangle ABC theta lim_theta rightarrow 0 R

Continuous Function In Exercises 101 and 102, find the value of c that makes the function continuous at x = 0. \(f(x)=\left\{\begin{array}{ll}\frac{4 x-2 \sin 2 x}{2 x^{3}}, & x \neq 0 \\c, & x=0\end{array}\right.\) Text Transcription: f(x)={4x-2 sin 2x/2x^3, x neq 0_c, x=0

Continuous Function In Exercises 101 and 102, find the value of c that makes the function continuous at x = 0. \(f(x)=\left\{\begin{array}{ll}\left(e^{x}+x\right)^{1 / x}, & x \neq 0 \\c, & x=0\end{array}\right.\) Text Transcription: f(x)={(e^x+x)^1/x, x neq 0_c, x=0

Finding Values Find the values of a and a such that \(\lim _{x \rightarrow 0} \frac{a-\cos b x}{x^{2}}=2\) Text Transcription: lim_x rightarrow 0 a-cos bx/x^2=2

Evaluating a Limit Use a graphing utility to graph \(f(x)=\frac{x^{k}-1}{k}\) for k = 1, 0.1, and 0.01. Then evaluate the limit \(\lim _{k \rightarrow 0^{+}} \frac{x^{k}-1}{k}\) Text Transcription: f(x)=x^k - 1/k lim_k rightarrow 0^+ x^k - 1/k

Finding a Derivative (a) Let f’(x) be continuous. Show that \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2 h}=f^{\prime}(x)\) (b) Explain the result of part (a) graphically. Text Transcription: lim_h rightarrow 0 f(x+h)-f(x-h)/2h = f’(x)

Finding a Second Derivative Let f’’(x) be continuous. Show that \(\lim _{h \rightarrow 0} \frac{f(x+h)-2 f(x)+f(x-h)}{h^{2}}=f^{\prime \prime}(x)\) Text Transcription: lim_h rightarrow 0 f(x+h)-2 f(x)+f(x-h)/h^2 = f’’(x)

Evaluating a Limit Consider the limit \(\lim _{x \rightarrow 0^{+}}(-x \ln x)\). (a) Describe the type of indeterminate form that is obtained by direct substitution. (b) Evaluate the limit. Use a graphing utility to verify the result. Text Transcription: lim_x rightarrow 0^+ (-x \ln x)

FOR FURTHER INFORMATION For a geometric approach to this exercise,see the article “A Geometric Proof of \(\lim _{d \rightarrow 0^{+}}(-d \ln d)=0\)” by John H. Mathews in The College Mathematics Journal.To view this article,go to MathArticles.com. Proof Prove that if \(f(x) \geq 0\), \(\lim _{x \rightarrow a} f(x)=0\), and \(\lim _{x \rightarrow a} g(x)=\infinity\), then \(\lim _{x \rightarrow a} f(x)^{g(x)}=0\) Text Transcription: lim_d rightarrow 0^+ (-d \ln d)=0 f(x) geq 0 lim_x rightarrow f(x)=0 lim_x rightarrow f(x)=infinity lim_x rightarrow f(x)^g(x) = 0

FOR FURTHER INFORMATION For a geometric approach to this exercise,see the article “A Geometric Proof of \(\lim _{d \rightarrow 0^{+}}(-d \ln d)=0\)” by John H. Mathews in The College Mathematics Journal.To view this article,go to MathArticles.com. Proof Prove that if \(f(x) \geq 0\), \(\lim _{x \rightarrow a} f(x)=0\), and \(\lim _{x \rightarrow a} g(x)=-\infinity\), then \(\lim _{x \rightarrow a} f(x)^{g(x)}=\infinity\) Text Transcription: lim_d rightarrow 0^+ (-d \ln d)=0 f(x) geq 0 lim_x rightarrow f(x)=0 lim_x rightarrow f(x)=-infinity lim_x rightarrow f(x)^g(x) = infinity

FOR FURTHER INFORMATION For a geometric approach to this exercise,see the article “A Geometric Proof of \(\lim _{d \rightarrow 0^{+}}(-d \ln d)=0\)” by John H. Mathews in The College Mathematics Journal.To view this article,go to MathArticles.com. Proof Prove the following generalization of the Mean Value Theorem. If f is twice differentiable on the closed interval [a, b], then \(f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t\). Text Transcription: lim_d rightarrow 0^+ (-d \ln d)=0 f(b)-f(a)=f’(a)(b-a)-int^b_a f’’(t)(t-b)dt.

Indeterminate Forms Show that the indeterminate forms \(0^{0}, \infty^{0} \text {, and } 1^{\infty}\) do not always have a value of 1 by evaluating each limit. (a) \(\lim _{x \rightarrow 0^{+}} x^{\ln 2 /(1+\ln x)}\) (b) \(\lim _{x \rightarrow \infinity} x^{\ln 2 /(1+\ln x)}\) (c) \(\lim _{x \rightarrow 0}(x+1)^{(\ln 2) / x}\) Text Transcription: 0^0, infinity^0, and 1^infinity lim_x rightarrow 0^+ x^ln 2/(1+ln x) lim_x rightarrow infinity x^ln 2/(1+ln x) lim_x rightarrow 0 (x+1)^(ln 2)/x

Calculus History In L’Hôpital’s 1696 calculus textbook, he illustrated his rule using the limit of the function \(f(x)=\frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}}{a-\sqrt[4]{a x^{3}}}\) as approaches a, a > 0. Find this limit. Text Transcription: f(x)=sqrt 2a^3x-x^4 -a 3 sqrt a^ax/a-4 sqrt ax^3

Finding a Limit Consider the function \(h(x)=\frac{x+\sin x}{x}\) (a) Use a graphing utility to graph the function. Then use the zoom and trace features to investigate \(\lim _{x \rightarrow \infty} h(x)\). (b) Find \(\lim _{x \rightarrow \infty} h(x)\) analytically by writing \(h(x)=\frac{x}{x}+\frac{\sin x}{x}\). (c) Can you use L’Hôpital’s Rule to find \(\lim _{x \rightarrow \infty} h(x)\)? Explain your reasoning. Text Transcription: h(x)x+sin x/x lim_x rightarrow infinity h(x) h(x)=x/x + sin x/x

Evaluating a Limit Let f(x) = x + x sin x and \(g(x)=x^{2}-4\). (a) Show that \(\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=0\). (b) Show that \(\lim _{x \rightarrow \infty} f(x)=\infty \text { and } \lim _{x \rightarrow \infty} g(x)=\infty\). (c) Evaluate the limit \(\lim _{x \rightarrow \infty} \frac{f^{\prime}(x)}{g^{\prime}(x)}\) What do you notice? (d) Do your answers to parts (a) through (c) contradict L’Hôpital’s Rule? Explain your reasoning. Text Transcription: g(x)=x^2-4 lim_x rightarrow infinity f(x)/g(x)=0 lim_x rightarrow infinity f(x)=infinity and lim_x rightarrow infinity g(x)=infinity lim_x rightarrow infinity f’(x)/g’(x)

PUTNAM EXAM CHALLENGE Evaluate \(\lim _{x \rightarrow \infty}\left[\frac{1}{x} \cdot \frac{a^{x}-1}{a-1}\right]^{1 / x} \text { where } a>0, a \neq 1\). Text Transcription: lim_x rightarrow infinity [1/x, a^x-1/a-a]^1/x where a > 0, a neq 1