a) Let be continuous. Show that (b) Explain the result of part (a) graphically

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. \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 3 x}\) 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. \(\lim _{x \rightarrow 0} \frac{1-e^{x}}{x}\) Text Transcription: lim_x rightarrow 0 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. \(\lim _{x \rightarrow \infty} x^{5} e^{-x / 100}\) Text Transcription: lim_x rightarrow infty 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. \(\lim _{x \rightarrow \infty} \frac{6 x}{\sqrt{3 x^{2}-2 x}}\) Text Transcription: lim_x rightarrow infty 6x / sqrt 3x^2 - 2x

In Exercises 5–10, evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L’Hôpital’s Rule. \(\lim _{x \rightarrow 4} \frac{3(x-4)}{x^{2}-16}\) Text Transcription: lim_x rightarrow 4 3(x-4) / x^2 - 16

In Exercises 5–10, evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L’Hôpital’s Rule. \(\lim _{x \rightarrow-2} \frac{2 x^{2}+x-6}{x+2}\) Text Transcription: lim_x rightarrow -2 2x^2 + x - 6 / x+2

In Exercises 5–10, evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L’Hôpital’s Rule. \(\lim _{x \rightarrow 6} \frac{\sqrt{x+10}-4}{x-6}\) Text Transcription: lim_x rightarrow 6 sqrt x+10 - 4 / x-6

In Exercises 5–10, evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L’Hôpital’s Rule. \(\lim _{x \rightarrow 0} \frac{\sin 6 x}{4 x}\) Text Transcription: lim_x rightarrow 0 sin 6x / 4x

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

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

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 3} \frac{x^{2}-2 x-3}{x-3}\) Text Transcription: lim_x rightarrow 3 x^2 - 2x - 3 / x-3

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow-1} \frac{x^{2}-2 x-3}{x+1}\) Text Transcription: lim_x rightarrow -1 x^2 - 2x - 3 / x+1

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 0} \frac{\sqrt{25-x^{2}}-5}{x}\) Text Transcription: lim_x rightarrow 0 sqrt 25-x^2 - 5 / x

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 5^{-}} \frac{\sqrt{25-x^{2}}}{x-5}\) Text Transcription: lim_x rightarrow 5^- sqrt 25-x^2 / x-5

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 0} \frac{e^{x}-(1-x)}{x}\) Text Transcription: lim_x rightarrow 0 e^x-(1-x) / x

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 1} \frac{\ln x^{2}}{x^{2}-1}\) Text Transcription: lim_x rightarrow 1 ln x^2 / x^2 - 1

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 0^{+}} \frac{e^{x}-(1+x)}{x^{3}}\) Text Transcription: lim_x rightarrow 0^+ e^x-(1+x) / x^3

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 0^{+}} \frac{e^{x}-(1+x)}{x^{n}}\) Text Transcription: lim_x rightarrow 0^+ e^x-(1+x) / x^n

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 1} \frac{x^{11}-1}{x^{4}-1}\) Text Transcription: lim_x rightarrow 1 x^11 - 1 / x^4 - 1

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 1} \frac{x^{a}-1}{x^{b}-1}\), where a, \(b \neq 0\) Text Transcription: lim_x rightarrow 1 x^a - 1 / x^b - 1 b neq 0

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 0} \frac{\sin 3 x}{\sin 5 x}\) Text Transcription: lim_x rightarrow 0 sin 3x / sin 5x

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}\), where a, \(b \neq 0\) Text Transcription: lim_x rightarrow 0 sin ax / sin bx b neq 0

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 0} \frac{\arcsin x}{x}\) Text Transcription: lim_x rightarrow 0 arcsin x / x

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 1} \frac{\arctan x-(\pi / 4)}{x-1}\) Text Transcription: lim_x rightarrow 1 arctan x-(pi/4) / x-1

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{5 x^{2}+3 x-1}{4 x^{2}+5}\) Text Transcription: lim_x rightarrow infty 5x^2 + 3x - 1 / 4x^2 + 5

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x-6}{x^{2}+4 x+7}\) Text Transcription: lim_x rightarrow infty x-6 / x^2+4x+7

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x^{2}+4 x+7}{x-6}\) Text Transcription: lim_x rightarrow infty x^2+4x+7 / x-6

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x^{3}}{x+2}\) Text Transcription: lim_x rightarrow infty x^3 / x+2

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{x / 2}}\) Text Transcription: lim_x rightarrow infty x^3 / e^x/2

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{x^{2}}}\) Text Transcription: lim_x rightarrow infty x^3 / e^x^2

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}}\) Text Transcription: lim_x rightarrow infty x / sqrt x^2 + 1

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x^{2}}{\sqrt{x^{2}+1}}\) Text Transcription: lim_x rightarrow infty x^2 / sqrt x^2 + 1

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{\cos x}{x}\) Text Transcription: lim_x rightarrow infty cos x / x

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{\sin x}{x-\pi}\) Text Transcription: lim_x rightarrow infty sin x / x-pi

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{\ln x}{x^{2}}\) Text Transcription: lim_x rightarrow infty ln x / x^2

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{\ln x^{4}}{x^{3}}\) Text Transcription: lim_x rightarrow infty ln x^4 / x^3

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{e^{x}}{x^{4}}\) Text Transcription: lim_x rightarrow infty e^x / x^4

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{e^{x / 2}}{x}\) Text Transcription: lim_x rightarrow infty e^x/2 / x

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 0} \frac{\sin 5 x}{\tan 9 x}\) Text Transcription: lim_x rightarrow 0 sin 5x / tan 9x

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 1} \frac{\ln x}{\sin \pi x}\) Text Transcription: lim_x rightarrow 1 ln x / sin pi x

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 0} \frac{\arctan x}{\sin x}\) Text Transcription: lim_x rightarrow 0 arctan x / sin x

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 0} \frac{x}{\arctan 2 x}\) Text Transcription: lim_x rightarrow 0 x / arctan 2x

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{\int_{1}^{x} \ln \left(e^{4 t-1}\right) \ d t}{x}\) Text Transcription: lim_x rightarrow infty int_1^x ln (e^4t-1) dt / x

In Exercises 11– 44, evaluate the limit, using L’Hôpital’s Rule if necessary. (In Exercise 18, n is a positive integer.) \(\lim _{x \rightarrow 1^{+}} \frac{\int_{1}^{x} \cos \theta \ d \theta}{x-1}\) Text Transcription: lim_x rightarrow 1^+ int_1^x cos theta d theta / x-1

In Exercises 45–62, (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). \(\lim _{x \rightarrow \infty} x \ \ln x\) Text Transcription: lim_x rightarrow infty x ln x

In Exercises 45–62, (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). \(\lim _{x \rightarrow 0^{+}} x^{3} \cot x\) Text Transcription: lim_x rightarrow 0^+ x^3 cot x

In Exercises 45–62, (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). \(\lim _{x \rightarrow \infty}\left(x \sin \frac{1}{x}\right)\) Text Transcription: lim_x rightarrow infty (x sin 1 / x)

In Exercises 45–62, (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). \(\lim _{x \rightarrow \infty} x \tan \frac{1}{x}\) Text Transcription: lim _x rightarrow infty x tan 1/x

In Exercises 45–62, (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). \(\lim _{x \rightarrow 0^{+}} x^{1 / x}\) Text Transcription: lim _x rightarrow 0^+ x^1 / x

In Exercises 45–62, (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). \(\lim _{x \rightarrow 0^{+}}\left(e^{x}+x\right)^{2 / x}\) Text Transcription: lim _x rightarrow 0^+ (e^x+x)^2 / x

In Exercises 45–62, (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 infty x^1 / x

In Exercises 45–62, (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 infty (1+1/x)^x

In Exercises 45–62, (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

In Exercises 45–62, (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 infty (1+x)^1/x

In Exercises 45–62, (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^+ [3x^x/2]

In Exercises 45–62, (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^+3x-4]^x-4

In Exercises 45–62, (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

In Exercises 45–62, (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^+left cos left frac pi 2-x right right^x

In Exercises 45–62, (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^+left frac 8 x^2-4-frac x x-2 right

In Exercises 45–62, (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^+ left frac1 x^2-4-frac sqrt x-1 x^2-4 right

In Exercises 45–62, (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^+left frac 3 ln x - frac 2 x-1 right

In Exercises 45–62, (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^+left frac 10 x-frac 3 x^2right

In Exercises 63–66, use a graphing utility to (a) graph the function and (b) find the required limit (if it exists). \(\lim _{x \rightarrow 3} \frac{x-3}{\ln (2 x-5)}\) Text Transcription: lim _x rightarrow 3 frac x-3 ln 2 x-5

In Exercises 63–66, use a graphing utility to (a) graph the function and (b) find the required limit (if it exists). \(\lim _{x \rightarrow 0^{+}}(\sin x)^{x}\) Text Transcription: lim _x rightarrow 0^+sin x^x

In Exercises 63–66, use a graphing utility to (a) graph the function and (b) find the required limit (if it exists). \(\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+5 x+2}-x\right)\) Text Transcription: lim _x rightarrow infty left sqrt x^2+5 x+2-x right

In Exercises 63–66, use a graphing utility to (a) graph the function and (b) find the required limit (if it exists). \(\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{2 x}}\) Text Transcription: lim _x rightarrow infty frac x^3 e^2 x

List six different indeterminate forms.

State L’Hôpital’s Rule.

Find differentiable functions f and g that satisfy the specified condition such that \(\lim _{x \rightarrow 5} f(x)=0\) and \(\lim _{x \rightarrow 5} g(x)=0\). 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 lim _{x \rightarrow 5} g(x)=0 lim _{x \rightarrow 5} \frac{f(x)}{g(x)}=10 lim _{x \rightarrow 5} \frac{f(x)}{g(x)}=0 lim _{x \rightarrow 5} \frac{f(x)}{g(x)}=\infty

Find differentiable functions and such that \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow \infty} g(x)=\infty\) 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 \infty} f(x)=\lim _{x \rightarrow \infty} g(x)=\infty \lim _{x \rightarrow \infty}[f(x)-g(x)]=25

Numerical Approach Complete the table to show that x eventually “overpowers” \((\ln x)^{4}\). Text Transcription: ln x^4

Numerical Approach Complete the table to show that \(e^{x}\) eventually “overpowers” \((\ln x)^{4}\). Text Transcription: e^x ln x^4

Comparing Functions In Exercises 73–78, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}\), \(g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{x^{2}}{e^{5 x}}\) Text Transcription: f(x)=x^m g(x)=e^n x h(x=ln x^n x rightarrow infty lim _x rightarrow infty frac x^2 e^5 x

Comparing Functions In Exercises 73–78, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}\), \(g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{2 x}}\) Text Transcription: f(x)=x^m g(x)=e^n x h(x=ln x^n x rightarrow infty lim _x rightarrow infty frac x^3 e^2 x

Comparing Functions In Exercises 73–78, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}\), \(g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{(\ln x)^{3}}{x}\) Text Transcription: f(x)=x^m g(x)=e^n x h(x=ln x^n x rightarrow infty lim _x rightarrow infty frac ln x^3 x

Comparing Functions In Exercises 73–78, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}\), \(g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{(\ln x)^{2}}{x^{3}}\) Text Transcription: f(x)=x^m g(x)=e^n x h(x=ln x^n x rightarrow infty lim _x rightarrow infty frac ln x^2 x^3

Comparing Functions In Exercises 73–78, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}\), \(g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{(\ln x)^{n}}{x^{m}}\) Text Transcription: f(x)=x^m g(x)=e^n x h(x=ln x^n x rightarrow infty lim _x rightarrow infty frac ln x^n x^m

Comparing Functions In Exercises 73–78, use L’Hôpital’s Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}\), \(g(x)=e^{n x}\), and \(h(x)=(\ln x)^{n}\), where n > 0, m > 0, \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{x^{m}}{e^{n x}}\) Text Transcription: f(x)=x^m g(x)=e^n x h(x=ln x^n x rightarrow infty lim _x rightarrow infty frac x^m e^n x

In Exercises 79–82, 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.) \(y=x^{1 / x}\), x > 0 Text Transcription: y=x^1 / x

In Exercises 79–82, 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.) \(y=x^{x}\), x > 0 Text Transcription: y=x^x

In Exercises 79–82, 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=2 x e^-x

In Exercises 79–82, 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=frac ln x x

Think About It In Exercises 83–87, L’Hôpital’s Rule is used incorrectly. Describe the error. Latex Transcription: \(\lim _{x \rightarrow 2} \frac{3 x^{2}+4 x+1}{x^{2}-x-2}=\lim _{x \rightarrow 2} \frac{6 x+4}{2 x-1}=\lim _{x \rightarrow 2} \frac{6}{2}=3\) Text Transcription: lim _x rightarrow 2 3 x^2+4 x+1 / x^2-x-2=lim _x rightarrow 2 6 x+4 / 2 x-1=lim _x rightarrow 2 6/2=3

Think About It In Exercises 83–87, L’Hôpital’s Rule is used incorrectly. Describe the error. Latex Transcription: \(\lim _{x \rightarrow 0} \frac{e^{2 x}-1}{e^{x}} \equiv \lim _{x \rightarrow 0} \frac{2 e^{2 x}}{e^{x}}=\lim _{x \rightarrow 0} 2 e^{x}=2\) Text Transcription: lim _x rightarrow 0 e^2 x-1 / e^x equiv lim _x rightarrow 0 2 e^{2 x / e^x=lim _x rightarrow 0 2 e^x=2

Think About It In Exercises 83–87, L’Hôpital’s Rule is used incorrectly. Describe the error. Latex Transcription: \(\lim _{x \rightarrow 0} \frac{\sin \pi x-1}{x} \equiv \lim _{x \rightarrow 0} \frac{\pi \cos \pi x}{1}=\pi\) Text Transcription: lim _x rightarrow 0 sin pi x-1 / x equiv lim _x rightarrow 0 pi cos pi x / 1=pi

Think About It In Exercises 83–87, L’Hôpital’s Rule is used incorrectly. Describe the error. Latex Transcription: \(\begin{aligned} \lim _{x \rightarrow \infty} x \cos \frac{1}{x} &=\lim _{x \rightarrow \infty} \frac{\cos (1 / x)}{1 / x} \\ &=\lim _{x \rightarrow \infty} \\ &=0 \end{aligned} \) Text Transcription: lim _x rightarrow infty x cos 1 x &=lim _x rightarrow infty cos 1 / x 1 / x = lim _x rightarrow infty =0

Think About It In Exercises 83–87, L’Hôpital’s Rule is used incorrectly. Describe the error. Latex Transcription: \(\lim _{x \rightarrow \infty} \frac{e^{-x}}{1+e^{-x}}=\lim _{x \rightarrow \infty} \frac{-e^{-x}}{-e^{-x}}\) \(\Rightarrow \lim _{x \rightarrow \infty}\) = 1 Text Transcription: lim _x rightarrow infty e^-x / 1+e^-x =lim _x rightarrow infty -e^-x / -e^-x Rightarrow lim _x rightarrow infty

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) \(\lim _{x \rightarrow 1} \frac{1+x(\ln x-1)}{\ln x(x-1)}\) Text Transcription: lim _x rightarrow 2 x-2 / x^3-x-6 lim _x rightarrow 0 x^2-4 x / 2 x-1 lim _x rightarrow infty 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 / ln x x-1

Analytical Approach In Exercises 89 and 90, (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 infty x / sqrt x^2+1

Analytical Approach In Exercises 89 and 90, (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 91 and 92, graph f(x)/g(x) and \(f^{\prime}(x) / g^{\prime}(x)\) near x = 0. What do you notice about these ratios as \(x \rightarrow 0\) How does this illustrate L’Hôpital’s Rule? \(f(x)=\sin 3 x\), gx = sin 4x Text Transcription: f^prime x / g^prime x x rightarrow 0 f(x)=\sin 3 x

Graphical Analysis In Exercises 91 and 92, graph f(x)/g(x) and \(f^{\prime}(x) / g^{\prime}(x)\) near x = 0. What do you notice about these ratios as \(x \rightarrow 0\) How does this illustrate L’Hôpital’s Rule? \(f(x)=e^{3 x}-1\), gx = x Text Transcription: f^prime x / g^prime x x rightarrow 0 fx=e^3x-1

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, t 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 k letting approach zero. (Assume that the downward direction is positive.) Text Transcription: v = 32 / k left(1-e^-k t+v_0 k e^-k t / 32 right v_0

Compound Interest The formula for the amount A in 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 left(1+r / n right^n t A=P e^r t

The Gamma Function The Gamma Function \(\Gamma(n)\) is defined in terms of the integral of the function given by \(f(x)=x^{n-1} e^{-x}\), 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) fx=x^n-1 e^-x

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=-sqrt144-x^2/x}

In Exercises 97–100, apply the Extended Mean Value Theorem to the functions f and g on the given interval. Find all values c in the interval (a, b) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\). Functions Interval \(f(x)=x^{3}, \quad g(x)=x^{2}+1\) [0, 1] Text Transcription: f^prime (c) / {g^prime(c)= f(b)-f(a)/g(b)-g(a) f(x)=x^3, quad g(x)=x^2+1

In Exercises 97–100, apply the Extended Mean Value Theorem to the functions f and g on the given interval. Find all values c in the interval (a, b) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\). Functions Interval \(f(x)=\frac{1}{x}, \quad g(x)=x^{2}-4\) [0, 2] Text Transcription: f^prime (c) / {g^prime(c)= f(b)-f(a)/g(b)-g(a) f(x)=1/x, quad g(x)=x^{2}-4

In Exercises 97–100, apply the Extended Mean Value Theorem to the functions f and g on the given interval. Find all values c in the interval (a, b) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\). Functions Interval \(f(x)=\sin x, \quad g(x)=\cos x\) \(\left[0, \frac{\pi}{2}\right]\) Text Transcription: f^prime (c) / {g^prime(c)= f(b)-f(a)/g(b)-g(a) f(x)=sin x, quad g(x) = cos x left 0, pi / 2 right

In Exercises 97–100, apply the Extended Mean Value Theorem to the functions f and g on the given interval. Find all values c in the interval (a, b) such that \(\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\). Functions Interval \(f(x)=\ln x, \quad g(x)=x^{3}\) [1, 4] Text Transcription: f^prime (c) / {g^prime(c)= f(b)-f(a)/g(b)-g(a) f(x)=ln x, quad g(x)=x^3

True or False? In Exercises 101–104, 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 [2 x+1 / 1=1]

True or False? In Exercises 101–104, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=e^{x} / x^{2}\), then \(y^{\prime}=e^{x} / 2 x\). Text Transcription: y=e^x / x^2 y^prime=e^x / 2 x

True or False? In Exercises 101–104, 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 \(\lim _{x \rightarrow \infty}\left[p(x) / e^{x}\right]=0\). Text Transcription: lim _x rightarrow infty [p(x) / e^x=0

True or False? In Exercises 101–104, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=1\), then \(\lim _{x \rightarrow \infty}[f(x)-g(x)]=0\). Text Transcription: lim _x rightarrow infty f(x) / g(x)=1 lim _{x rightarrow infty [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.

In Section 1.3, a geometric argument (see figure) was used to prove that \(\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1\) (a) Write the area of \(\triangle A B D\) in terms of \(\theta\). (b) Write the area of the shaded region in terms of \(\theta\). (c) Write the ratio R of the area of \(\triangle A B D\) to that of the shaded region. (d) Find \(\lim _{\theta \rightarrow 0} R\). Text Transcription: lim _theta rightarrow 0 sin theta / theta=1 triangle A B D theta lim _theta rightarrow 0 R

Continuous Functions In Exercises 107 and 108, 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) = {4 x-2 sin 2 x / 2 x^3, & x neq 0 c, & x=0

Continuous Functions In Exercises 107 and 108, 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

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

Show that \(\lim _{x \rightarrow \infty} \frac{x^{n}}{e^{x}}=0\) for any integer n > 0. Text Transcription: lim _x rightarrow infty x^n / e^x=0

(a) Let \(f^{\prime}(x)\) 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: f^prime x lim _h rightarrow 0 f x+h - f x-h / 2 h=f^prime x

Let \(f^{\prime \prime}(x)\) 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: f^{\prime \prime}(x) lim _h rightarrow 0 fx+h - 2 fx+fx-h / h^2 = f^prime prime x

Sketch the graph of \(g(x)=\left\{\begin{array}{ll} e^{-1 / x^{2}}, & x \neq 0 \\ 0, & x=0 \end{array}\right. \) and determine \(g^{\prime}(0)\). Text Transcription: g(x)=e^{-1 / x^2, & x neq 0 0, & x=0 g^prime 0

Use a graphing utility to graph \(f(x)=\frac{x^{k}-1}{k}\) for 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

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

Prove that if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0, \text { and } \lim _{x \rightarrow a} g(x)=\infty\), then \(\lim _{x \rightarrow a} f(x)^{g(x)}=0\). Text Transcription: f(x) geq 0, lim _x rightarrow a f(x)=0 lim _x rightarrow a g(x) = infty lim _x rightarrow a f(x)^g(x)=0

Prove that if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0, \text { and } \lim _{x \rightarrow a} g(x)=-\infty\), then \(\lim _{x \rightarrow a} f(x)^{g(x)}=\infty\). Text Transcription: f(x) geq 0, lim _{x \rightarrow a} f(x)=0 lim _x rightarrow a g(x)=-infty lim _x rightarrow a f(x)^g(x) = infty

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: f(b)-f(a)=f^prime a b-a -int_a^b f^prime prime t t-b d t

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) \(\lim _{x \rightarrow \infty} x^{\ln 2 /(1+\ln x)}\) (c) \(\lim _{x \rightarrow 0}(x+1)^{(\ln 2) / x}\) Text Transcription: 0^0 infty^0 1^infty lim _x rightarrow 0^+x^ln 2 1+\ln x lim _x rightarrow infty x^ln 2 1+ln x lim _x rightarrow 0 x+1^ln 2 x

120. 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 x approaches a, a > 0 Find this limit. Text Transcription: f(x) = sqrt 2 a^3 x-x^4-a sqrt 3 a^2 x / {a-sqrt 4 a x^3

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 infty h(x) h(x) = x / x+sin x / x

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 infty f(x) / g(x)=0 lim _x rightarrow infty f(x)=\infty lim _x rightarrow infty g(x)=\infty

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