91Ó°ÊÓ

\(\begin{array}{llll}\text { Comparing } & \text { Functions } & \text { In Exercises } & \mathbf{4 5}-\mathbf{5 0}, & \text { use } & \text { L'Hôpital's }\end{array}\) 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 \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{x^{m}}{e^{n x}}\)

Find the integral. Use a computer algebra system to confirm your result. $$ \int \frac{1-\sec t}{\cos t-1} d t $$

In Exercises \(51-54,\) 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.) \(y=x^{1 / x}, \quad x>0\)

Evaluate the definite integral. $$ \int_{0}^{\pi / 4} \tan ^{3} x d x $$

Evaluate the definite integral. $$ \int_{0}^{\pi / 4} \sec ^{2} t \sqrt{\tan t} d t $$

Find the volume of the torus generated by revolving the region bounded by the graph of the circle about the \(y\) -axis. $$ (x-h)^{2}+y^{2}=r^{2}, h>r $$

In Exercises \(51-54,\) 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.) \(y=x^{x}, \quad x>0\)

Find the arc length of the curve over the given interval. $$ y=\ln x, \quad[1,5] $$

In Exercises \(51-54,\) 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.) \(y=2 x e^{-x}\)

Integrate \(\int \frac{x^{3}}{\sqrt{4+x^{2}}} d x\) (a) by parts, letting \(d v=\left(x / \sqrt{4+x^{2}}\right) d x\). (b) by substitution, letting \(u=4+x^{2}\).