Chapter 8

In Problems 1-24, evaluate each improper integral or show that it diverges. \(\int_{100}^{\infty} e^{x} d x\)

In Problems \(1-32\), evaluate each improper integral or show that it diverges. $$ \text { 1. } \int_{1}^{3} \frac{d x}{(x-1)^{1 / 3}} $$

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} \frac{\ln x^{10000}}{x} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{2 x-\sin x}{x} $$

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{-5} \frac{d x}{x^{4}}\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} \frac{(\ln x)^{2}}{2^{x}} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow \pi / 2} \frac{\cos x}{\frac{1}{2} \pi-x} $$

Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} 2 x e^{-x^{2}} d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} \frac{x^{10000}}{e^{x}} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{x-\sin 2 x}{\tan x} $$

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{1} e^{4 x} d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} \frac{3 x}{\ln \left(100 x+e^{x}\right)} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\tan ^{-1} 3 x}{\sin ^{-1} x} $$

Evaluate each improper integral or show that it diverges. \(\int_{9}^{\infty} \frac{x d x}{\sqrt{1+x^{2}}}\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \pi / 2} \frac{3 \sec x+5}{\tan x} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow-2} \frac{x^{2}+6 x+8}{x^{2}-3 x-10} $$

Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \frac{d x}{\sqrt{\pi x}}\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{\ln \sin ^{2} x}{3 \ln \tan x} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{x^{3}-3 x^{2}+x}{x^{3}-2 x} $$

Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \frac{d x}{x^{1.00001}}\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} \frac{\ln \left(\ln x^{1000}\right)}{\ln x} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 1^{-}} \frac{x^{2}-2 x+2}{x^{2}-1} $$

Evaluate each improper integral or show that it diverges. \(\int_{10}^{\infty} \frac{x}{1+x^{2}} d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow(1 / 2)^{-}} \frac{\ln (4-8 x)^{2}}{\tan \pi x} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 1} \frac{\ln x^{2}}{x^{2}-1} $$

Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \frac{d x}{x^{0.99999}}\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{\cot x}{\sqrt{-\ln x}} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow \pi / 2} \frac{\ln (\sin x)^{3}}{\frac{1}{2} \pi-x} $$

Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \frac{x}{\left(1+x^{2}\right)^{2}} d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{2 \csc ^{2} x}{\cot ^{2} x} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{2 \sin x} $$

Evaluate each improper integral or show that it diverges. \(\int_{e}^{\infty} \frac{1}{x \ln x} d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0}\left(x \ln x^{1000}\right) $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{t \rightarrow 1} \frac{\sqrt{t}-t^{2}}{\ln t} $$

Evaluate each improper integral or show that it diverges. \(\int_{e}^{\infty} \frac{\ln x}{x} d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} 3 x^{2} \csc ^{2} x $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{7^{\sqrt{x}}-1}{2^{\sqrt{x}}-1} $$

Evaluate each improper integral or show that it diverges. \(\int_{2}^{\infty} \frac{\ln x}{x^{2}} d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0}\left(\csc ^{2} x-\cot ^{2} x\right) $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\ln \cos 2 x}{7 x^{2}} $$

Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} x e^{-x} d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \pi / 2}(\tan x-\sec x) $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{-}} \frac{3 \sin x}{\sqrt{-x}} $$

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{1} \frac{d x}{(2 x-3)^{3}}\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}}(3 x)^{x^{2}} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\tan x-x}{\sin 2 x-2 x} $$

Evaluate each improper integral or show that it diverges. \(\int_{4}^{\infty} \frac{d x}{(\pi-x)^{2 / 3}}\)

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\sin x-\tan x}{x^{2} \sin x} $$

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{\infty} \frac{x}{\sqrt{x^{2}+9}} d x\)

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{x^{2}}{\sin x-x} $$