91Ó°ÊÓ

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0} \frac{1}{x^{2}-1} $$

In Problems 15-24, find the values of \(x \in\) R for which the given functions are both defined and continuous. $$ f(x)=\cos (2 x) $$

Let \(f(x)=x^{3} \cos \frac{1}{x}, \quad x \neq 0\) (a) Use a graphing calculator to sketch the graph of \(y=f(x)\). (b) Use the sandwich theorem to show that \(\lim _{x \rightarrow 0} x^{3} \cos \frac{1}{x}=0\).

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 3^{+}} \frac{1}{3-x}=-\infty $$

Evaluate the limits. $$ \lim _{x \rightarrow \infty} \frac{3 e^{2 x}+1}{2 e^{2 x}-e^{x}} $$

Let \(f(x)=\frac{\ln x}{x}, \quad x>0\) (a) Use a graphing calculator to graph \(y=f(x)\). (b) Use a graphing calculator to investigate the values of \(x\) for which $$\frac{1}{x} \leq \frac{\ln x}{x} \leq \frac{1}{\sqrt{x}}$$ holds. (c) Use your result in (b) to explain why: \(\lim _{x \rightarrow \infty} \frac{\ln x}{x}=0\).

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0^{+}} \frac{1}{\left(1-e^{-x}\right)} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 0^{-}} \frac{-4}{x}=\infty. $$

Evaluate the limits. $$ \lim _{x \rightarrow \infty} \frac{3 e^{2 x}}{2 e^{2 x}-e^{x}} $$

Let \(f(x)=\frac{\sin ^{2} x}{x}, \quad x>0\) (a) Use a graphing calculator to graph \(y=f(x)\). (b) Explain why you cannot use the basic rules for finding limits to compute \(\lim _{x \rightarrow \infty} \frac{\sin ^{2} x}{x}\) (c) Show that \(0 \leq \frac{\sin ^{2} x}{x} \leq \frac{1}{x}\) holds for \(x>0\), and use the sandwich theorem to compute \(\lim _{x \rightarrow \infty} \frac{\sin ^{2} x}{x}\)