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Magazine Problem 7
Find each limit. Be sure you have an indeterminate form before applying l'H么pital's Rule. $$ \lim _{x \rightarrow \infty} \frac{\ln \left(\ln x^{1000}\right)}{\ln x} $$
Problem 7
Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \frac{d x}{x^{1.00001}}\)
Problem 8
Find each limit. Be sure you have an indeterminate form before applying l'H么pital's Rule. $$ \lim _{x \rightarrow(1 / 2)^{-}} \frac{\ln (4-8 x)^{2}}{\tan \pi x} $$
Problem 8
Evaluate each improper integral or show that it diverges. \(\int_{10}^{\infty} \frac{x}{1+x^{2}} d x\)
Problem 8
Find the indicated limit. Make sure that you have an indeterminate form before you apply l'H么pital's Rule. $$ \lim _{x \rightarrow 1} \frac{\ln x^{2}}{x^{2}-1} $$
Problem 9
Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \frac{d x}{x^{0.99999}}\)
Problem 9
Find the indicated limit. Make sure that you have an indeterminate form before you apply l'H么pital's Rule. $$ \lim _{x \rightarrow \pi / 2} \frac{\ln (\sin x)^{3}}{\frac{1}{2} \pi-x} $$
Problem 9
Find each limit. Be sure you have an indeterminate form before applying l'H么pital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{\cot x}{\sqrt{-\ln x}} $$
Problem 10
Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \frac{x}{\left(1+x^{2}\right)^{2}} d x\)
Problem 10
Find the indicated limit. Make sure that you have an indeterminate form before you apply l'H么pital's Rule. $$ \lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{2 \sin x} $$
