91Ó°ÊÓ

Find the limit of the sequence. $$\lim _{x \rightarrow \infty}(\sqrt{n}-1)^{1 / \sqrt{n}}$$

Let \(M\) be a positive integer. Show that \(a_{n}=M^{n} / n !\) decreases for \(n \geq M\).

Show that if \(\quad 0

Find the fallacy: $$\lim _{x \rightarrow 0} \frac{2+x+\sin x}{x^{3}+x-\cos x} \div \lim _{x \rightarrow 0} \frac{1+\cos x}{3 x^{2}+1+\sin x}$$ $$\doteq \lim _{x \rightarrow 0} \frac{-\sin x}{6 x+\cos x}=\frac{0}{1}=0$$

Show that, if \(0

Let \(\Omega\) be the region hounded by the coordinate axes. the curve \(y = 1 / \sqrt{x}\), and the line \(x=1\). (a) Sketch \(\Omega\). (b) Show that 2 has finite area and find it. (c) Show that if \(\Omega\) is revolved about the \(x\) -axis. the configuration obtained does not have finite volume.

Find the indicated limit.$$\begin{aligned}&\lim _{n \rightarrow \infty} \frac{1+2+\cdots+n}{n^{2}}\\\ &\text { HINT: } 1+2+\cdots+n=\frac{n(n+1)}{2}\end{aligned}$$.

Show that, if \(a > 0\), then $$\lim _{n \rightarrow \infty} n\left(a^{1 / n}-1\right)=\ln a$$

Use technology (graphing utility or CAS) to calculate the limit. $$\lim _{x \rightarrow\left(\frac{\pi}{2}\right)^{-}}(\tan x)^{\tan 2 x}$$

Let \(\Omega\) be the region between the curve \(y=1 /\left(1+x^{2}\right)\) and the \(x\) -axis, \(x \geq 0\). (a) Sketch \(\Omega\). (b) Find the area of \(\Omega\). (c) Find the volume obtained by revolving \(\Omega\) about the \(x\)-axis. (d) Find the volume obtained by revolving \(\Omega\) about the \(y\) -axis.