Chapter 8

Evaluate each improper integral or show that it diverges. \(\int_{-\infty}^{\infty} \frac{d x}{\left(x^{2}+16\right)^{2}}\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0}\left(\csc ^{2} x-\frac{1}{x^{2}}\right)^{2} $$

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

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

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

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\tan ^{-1} x-x}{8 x^{3}} $$

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

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

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

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

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

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

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

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

Evaluate each improper integral or show that it diverges. \(\int_{0}^{\infty} e^{-x} \cos x d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} x^{1 / x} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{\int_{0}^{x} \sqrt{1+\sin t} d t}{x} $$

Evaluate each improper integral or show that it diverges. \(\int_{0}^{\infty} e^{-x} \sin x d x\)

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0}(\cos x)^{1 / x^{2}} $$

Find the indicated limit. Make sure that you have an indeterminate form before you apply l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{t} \cos t d t}{x^{2}} $$

Evaluate each improper integral or show that it diverges. $$ \int_{0}^{\pi} \frac{d x}{\cos x-1} $$

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

Find the area of the region under the curve \(y=\) \(1 /\left(x^{2}+x\right)\) to the right of \(x=1\)

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

Find \(\lim _{x \rightarrow 0} \frac{x^{2} \sin (1 / x)}{\tan x}\).

Suppose that Newton's law for the force of gravity had the form \(-k / x\) rather than \(-k / x^{2}\) (see Example 3). Show that it would then be impossible to send anything out of the earth's gravitational field.

Evaluate each improper integral or show that it diverges. $$ \int_{0}^{\ln 3} \frac{e^{x} d x}{\sqrt{e^{x}-1}} $$

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

Evaluate each improper integral or show that it diverges. $$ \int_{2}^{4} \frac{d x}{\sqrt{4 x-x^{2}}} $$

Suppose that a company expects its annual profits \(t\) years from now to be \(f(t)\) dollars and that interest is considered to be compounded continuously at an annual rate \(r\). Then the present value of all future profits can be shown to be $$ F P=\int_{0}^{\infty} e^{-r t} f(t) d t $$ Find \(F P\) if \(r=0.08\) and \(f(t)=100,000\).

Evaluate each improper integral or show that it diverges. $$ \int_{1}^{2} \frac{d x}{x \ln x} $$

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0}\left(\csc x-\frac{1}{x}\right) $$

Let $$f(x)=\left\\{\begin{array}{ll} \frac{e^{x}-1}{x}, & \text { if } x \neq 0 \\ c, & \text { if } x=0 \end{array}\right.$$ What value of \(c\) makes \(f(x)\) continuous at \(x=0\) ?

Evaluate each improper integral or show that it diverges. $$ \int_{1}^{10} \frac{d x}{x \ln ^{100} x} $$

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x} $$

Let $$f(x)=\left\\{\begin{array}{ll} \frac{\ln x}{x-1}, & \text { if } x \neq 1 \\ c, & \text { if } x=1 \end{array}\right. $$ What value of \(c\) makes \(f(x)\) continuous at \(x=1\) ?

A continuous random variable \(X\) has a uniform distribution if it has a probability density function of the form $$ f(x)=\left\\{\begin{array}{ll} \frac{1}{b-a} & \text { if } a

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}}\left(1+2 e^{x}\right)^{1 / x} $$

A random variable \(X\) has a Weibull distribution if it has probability density function $$ f(x)=\left\\{\begin{array}{ll} \frac{\beta}{\theta}\left(\frac{x}{\theta}\right)^{\beta-1} e^{-(x / \theta)^{\beta}} & \text { if } x>0 \\ 0 & \text { if } x \leq 0 \end{array}\right. $$ (a) Show that \(\int_{-\infty}^{\infty} f(x) d x=1\). (Assume \(\beta>1\).) (b) If \(\theta=3\) and \(\beta=2\), find the mean \(\mu\) and the variance \(\sigma^{2}\). (c) If the lifetime of a computer monitor is a random variable \(X\) that has a Weibull distribution with \(\theta=3\) and \(\beta=2\) (where age is measured in years) find the probability that a monitor fails before two years.

Evaluate each improper integral or show that it diverges. $$ \int_{c}^{2 c} \frac{x d x}{\sqrt{x^{2}+x c-2 c^{2}}}, c>0 $$

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 1}\left(\frac{1}{x-1}-\frac{x}{\ln x}\right) $$

Sketch the graph of the normal probability density function $$ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} / 2 \sigma^{2}} $$ and show, using calculus, that \(\sigma\) is the distance from the mean \(\mu\) to the \(x\) -coordinate of one of the inflection points.

It is often possible to change an improper integral into a proper one by using integration by parts. Consider \(\lim _{c \rightarrow 0^{+}} \int_{c}^{1} \frac{d x}{\sqrt{x}(1+x)} .\) Use integration by parts on the interval \([c, 1]\) where \(0

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0}(\cos x)^{1 / x} $$

L'Hôpital's Rule in its 1696 form said this: If \(\lim _{x \rightarrow a} f(x)=\lim _{x \rightarrow a} g(x)=0\), then \(\lim _{x \rightarrow a} f(x) / g(x)=f^{\prime}(a) / g^{\prime}(a)\) provided that \(f^{\prime}(a)\) and \(g^{\prime}(a)\) both exist and \(g^{\prime}(a) \neq 0\). Prove this result without recourse to Cauchy's Mean Value Theorem.

The Pareto probability density function has the form $$ f(x)=\left\\{\begin{array}{ll} \frac{C M^{k}}{x^{k+1}} & \text { if } x \geq M \\ 0 & \text { if } x

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0^{+}}\left(x^{1 / 2} \ln x\right) $$

The Pareto distribution is often used to model income distribution. Suppose that in some economy the income distribution does follow a Pareto distribution with \(k=3 .\) Suppose that the mean income is $$\$ 20,000$$. (a) Find \(M\) and \(C\). (b) Find the variance \(\sigma^{2}\). (c) Find the fraction of income earners who earn more than $$\$ 100,000$$. (Note: This is the same as asking what is the probability that a randomly chosen person has an income of more than $$\$ 100,000 .$$ )

If \(f(x)\) tends to infinity at both \(a\) and \(b\), then we define $$ \int_{a}^{b} f(x) d x=\int_{a}^{c} f(x) d x+\int_{c}^{b} f(x) d x $$ where \(c\) is any point between \(a\) and \(b\), provided of course that both the latter integrals converge. Otherwise, we say that the given integral diverges. Use this to evaluate \(\int_{-3}^{3} \frac{x}{\sqrt{9-x^{2}}} d x\) or show that it diverges.

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