91Ó°ÊÓ

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

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

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

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

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

A random variable \(X\) has a Weibull distribution if it has probability density function $$ f(x)= \begin{cases}\frac{\beta}{\theta}\left(\frac{x}{\theta}\right)^{\beta-1} e^{-(x / \theta)^{\beta}} & \text { if } x>0 \\ 0 & \text { if } x \leq 0\end{cases} $$ (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.

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

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

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