91Ó°ÊÓ

(a) Use a graphing utility to generate the graph of the function \(f(x)=x /\left(x^{3}-x+2\right)\), and then use the graph to make a conjecture about the number and locations of all discontinuities. (b) Use the Intermediate-Value Theorem to approximate the locations of all discontinuities to two decimal places.

First rationalize the numerator and then find the limit. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+4}-2}{x} $$

(a) Find the smallest positive number \(N\) such that for each \(x\) in the interval \((N,+\infty)\), the value of the function \(f(x)=1 / x^{2}\) is within \(0.1\) unit of \(L=0\) (b) Find the smallest positive number \(N\) such that for each \(x\) in the interval \((N,+\infty)\), the value of \(f(x)=x /(x+1)\) is within \(0.01\) unit of \(L=1\). (c) Find the largest negative number \(N\) such that for each \(x\) in the interval \((-\infty, N)\), the value of the function \(f(x)=1 / x^{3}\) is within \(0.001\) unit of \(L=0\) (d) Find the largest negative number \(N\) such that for each \(x\) in the interval \((-\infty, N)\), the value of the function \(f(x)=x /(x+1)\) is within \(0.01\) unit of \(L=1\)

Find the limits. $$ \lim _{x \rightarrow+\infty} \frac{(x+1)^{x}}{x^{x}} $$

Let $$f(x)=\frac{x^{3}-1}{x-1}$$ (a) Find \(\lim _{x \rightarrow 1} f(x)\). (b) Sketch the graph of \(y=f(x)\).

Prove that \(f(x)=x^{3 / 5}\) is continuous everywhere, carefully justifying each step.

Find the limits. $$ \lim _{x \rightarrow 0} \frac{2-\cos 3 x-\cos 4 x}{x} $$

Prove that \(f(x)=1 / \sqrt{x^{4}+7 x^{2}+1}\) is continuous everywhere, carefully justifying each step.

Find the limits. $$ \lim _{x \rightarrow 0} \frac{\tan 3 x^{2}+\sin ^{2} 5 x}{x^{2}} $$

Find the limits. $$ \lim _{x \rightarrow+\infty}\left(1+\frac{1}{x}\right)^{-x} $$