Chapter 3

Let $$f(x)=x^{2} \cos \frac{1}{x}, \quad x \neq 0$$ (a) Use a graphing calculator to sketch the graph of \(y=\overline{f(x) \text { . }}\) (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 $$ \lim _{x \rightarrow 0} x^{2} \cos \frac{1}{x}=0 $$

Find the values of \(x\) such that $$ |2 x-1|<0.01 $$

Let $$ f(x)=x^{2}-1, \quad 0 \leq x \leq 2 $$ (a) Graph \(y=f(x)\) for \(0 \leq x \leq 2\). (b) Show that $$ f(0)<0

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 2}\left(x^{2}-4 x+1\right) $$

In Problems 1-4, show that each function is continuous at the given value. $$ f(x)=2 x, c=1 / 2 $$

Evaluate the limits in problems. $$ \lim _{x \rightarrow \infty} \frac{2 x^{2}-3 x+5}{x^{4}-2 x+1} $$

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

Find the values of \(x\) such that $$ |3 x-9|<0.01 $$

Let $$ f(x)=x^{3}-2 x+3, \quad-3 \leq x \leq-1 $$ (a) Graph \(y=f(x)\) for \(-3 \leq x \leq-1\). (b) Use the intermediate-value theorem to conclude that $$ x^{3}-2 x+3=0 $$ has a solution in \((-3,-1)\).

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

In Problems 1-4, show that each function is continuous at the given value. $$ f(x)=-x, c=1 $$

Evaluate the limits in problems. $$ \lim _{x \rightarrow \infty} \frac{x^{2}+3}{5 x^{2}-2 x+1} $$

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 the following is true: $$ \lim _{x \rightarrow \infty} \frac{\ln x}{x}=0 $$

Find the values of \(x\) such that $$ \left|x^{2}-9\right|<0.1 $$

Let $$ f(x)=\sqrt{x^{2}+2}, \quad 1 \leq x \leq 2 $$ (a) Graph \(y=f(x)\) for \(1 \leq x \leq 2\). (b) Use the intermediate-value theorem to conclude that $$ \sqrt{x^{2}+2}=2 $$ has a solution in \((1,2)\).

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

In Problems 1-4, show that each function is continuous at the given value. $$ f(x)=x^{3}-2 x+1, c=2 $$

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

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

Find the values of \(x\) such that $$ |2 \sqrt{x}-5|<0.1 $$

Let $$ f(x)=\sin x-x, \quad-1 \leq x \leq 1 $$ (a) Graph \(y=f(x)\) for \(-1 \leq x \leq 1\). (b) Use the intermediate-value theorem to conclude that $$ \sin x=x $$ has a solution in \((-1,1)\).

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{s \rightarrow 2} s\left(s^{2}-4\right) $$

In Problems 1-4, show that each function is continuous at the given value. $$ f(x)=x^{2}+1, c=-1 $$

Evaluate the limits in problems. $$ \lim _{x \rightarrow-\infty} \frac{2 x-1}{3-4 x} $$

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

Let $$ f(x)=2 x-1, x \in \mathbf{R} $$ (a) Graph \(y=f(x)\) for \(-3 \leq x \leq 5\). (b) For which values of \(x\) is \(y=f(x)\) within \(0.1\) of 3 ? [Hint: Find values of \(x\) such that \(|(2 x-1)-3|<0.1\).] (c) Illustrate your result in (b) on the graph that you obtained in (a).

Use the intermediate-value theorem to show that $$ e^{-x}=x $$ has a solution in \((0,1)\).

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

Show that $$ f(x)=\left\\{\begin{array}{cl} \frac{x^{2}-x-2}{x-2} & \text { if } x \neq 2 \\ 3 & \text { if } x=2 \end{array}\right. $$ is continuous at \(x=2\).

Evaluate the limits in problems. $$ \lim _{x \rightarrow \infty} \frac{1-x^{3}+2 x^{4}}{2 x^{2}+x^{4}} $$

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{\sin (2 x)}{3 x} $$

Let $$ f(x)=\sqrt{x}, \quad x \geq 0 $$ (a) Graph \(y=f(x)\) for \(0 \leq x \leq 6\). (b) For which values of \(x\) is \(y=f(x)\) within \(0.2\) of \(1 ?\) (Hint: Find values of \(x\) such that \(|\sqrt{x}-1|<0.2\).) (c) Illustrate your result in (b) on the graph that you obtained in (a).

Use the intermediate-value theorem to show that $$ \cos x=x $$ has a solution in \((0,1)\).

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{t \rightarrow \pi / 9} \sin (3 t) $$

Evaluate the limits in problems. $$ \lim _{x \rightarrow \infty} \frac{1-5 x^{3}}{1+3 x^{4}} $$

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{\sin (5 x)}{x} $$

Let $$ f(x)=\frac{1}{x}, \quad x>0 $$ (a) Graph \(y=f(x)\) for \(0

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

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

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

Let $$ f(x)=e^{-x}, \quad x \geq 0 $$ (a) Graph \(y=f(x)\) for \(0 \leq x \leq 6\). (b) For which values of \(x\) is \(y=f(x)\) less than 0.1? (c) Illustrate your result in (b) on the graph that you obtained in (a).

Let $$ f(x)=\left\\{\begin{array}{cc} \frac{x^{2}+x-2}{x-1} & \text { if } x \neq 1 \\ a & \text { if } x=1 \end{array}\right. $$ Which value must you assign to \(a\) so that \(f(x)\) is continuous at \(x=1 ?\)

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

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{\sin (\pi x)}{x} $$

In Problems 9-12, determine at which points \(f(x)\) is discontinuous. $$ f(x)=\frac{1}{x-3} $$

Evaluate the limits in problems. $$ \lim _{x \rightarrow-\infty} \frac{x^{2}-3 x+1}{4-x} $$

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

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

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0} \frac{e^{x}+1}{2 x+3} $$

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