Chapter 3

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x^{2}} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 0} 2 x^{3}=0. $$

In Problems 9-12, determine at which points \(f(x)\) is discontinuous. $$ f(x)=\frac{1}{x^{2}-1} $$

Evaluate the limits. $$ \lim _{x \rightarrow-\infty} \frac{1+x^{4}}{2+x^{2}} $$

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos 2 x}{x} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 0} x^{5}=0. $$

Explain why a polynomial of degree 3 has at least one root.

In Problems 9-12, determine at which points \(f(x)\) is discontinuous. $$ f(x)=\left\\{\begin{array}{cl} x^{2}-1 & \text { if }|x| \geq 1 \\ x-1 & \text { if }|x|<1 \end{array}\right. $$

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

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

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

In Problems 9-12, determine at which points \(f(x)\) is discontinuous. $$ f(x)=\left\\{\begin{array}{cl} x^{2}-1 & \text { if } x \leq 0 \\ x-1 & \text { if } x>0 \end{array}\right. $$

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

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

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos (5 x)}{2 x} $$

Explain why \(y=x^{2}-5\) has at least two roots.

Evaluate the limits. $$ \lim _{x \rightarrow \infty} \frac{4 e^{-x}}{1+e^{-2 x}} $$

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 3} \frac{x^{2}-16}{x-4} $$

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos (x / 2)}{x} $$

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

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 3} \frac{x^{2}-9}{x+3} $$

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{\sin x(1-\cos x)}{x^{2}} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 0} \frac{1}{x^{4}}=\infty. $$

(a) Use the Intermediate Value Theorem to show that \(e^{x}=2-x\) has a solution in \((0,2) .\) (b) Find this solution to an accuracy of \(10^{-4}\) using the bisection search method, implemented as a spreadsheet.

In Problems 15-24, find the values of \(x \in\) R for which the given functions are both defined and continuous. $$ f(x)=3 x^{4}-x^{2}+4 $$

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

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow \pi / 2} \sin (2 x) $$

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 3} \frac{1}{(x-3)^{2}}=\infty. $$

Use the Intermediate Value Theorem to show that \(e^{x}-\) \(e^{-1.5 x}-1=0\) has a solution in the interval \((0,1) .\) Use a spreadsheet to calculate the value of this root to an accuracy of \(10^{-5}\).

In Problems 15-24, find the values of \(x \in\) R for which the given functions are both defined and continuous. $$ f(x)=\sqrt{x^{2}-1} $$

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

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow \pi / 2} \cos (x-\pi) $$

17\. LL(a) Use a graphing calculator to sketch the graph of \(y=f(x)\). (b) Show that \(-x^{2} \leq x^{2} \cos \frac{1}{x} \leq x^{2}\) holds for \(x \neq 0\). (c) Use your result in (b) and the sandwich theorem to show that \(\lim _{x \rightarrow 0} x^{2} \cos \frac{1}{x}=0\).

For each of the following equations show that the equation has a \mathrm{\\{} \text { root in the given interval. Then use the bisection search method, } implemented as a spreadsheet, to find this root to an accuracy of \(10^{-5}\). \(x^{3}-2 x+1=0 \quad(-2,-1)\)

In Problems 15-24, find the values of \(x \in\) R for which the given functions are both defined and continuous. $$ f(x)=\frac{x^{2}+1}{x-1} $$

Evaluate the limits. $$ \lim _{x \rightarrow-\infty} \exp [x] $$

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

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 \(\lim _{x \rightarrow 0} x^{3} \cos \frac{1}{x}=0\).

For each of the following equations show that the equation has a \mathrm{\\{} \text { root in the given interval. Then use the bisection search method, } implemented as a spreadsheet, to find this root to an accuracy of \(10^{-5}\). \(x^{5}+x^{2}-x+1=0 \quad(-2,-1)\)

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

Evaluate the limits. $$ \lim _{x \rightarrow \infty} \exp \left[-x^{2}\right] $$

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

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: \(\lim _{x \rightarrow \infty} \frac{\ln x}{x}=0\).

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

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

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

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 \(\lim _{x \rightarrow \infty} \frac{\sin ^{2} x}{x}\)

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

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