Chapter 3

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

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

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

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

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

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{\sin (2 x)}{4 x} $$

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

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

In Problems \(1-4\), show that each function is continuous at the given value. $$ f(x)=-x, c=0 $$

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

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

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

Let \(f(x)=\sqrt{x}+x, \quad 1 \leq x \leq 2\). (a) Graph \(y=f(x)\) for \(1 \leq x \leq 2\). (b) Use the Intermediate Value Theorem to conclude that \(\sqrt{x}+x=3\) has a solution in \((1,2)\).

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. $$ \lim _{x \rightarrow-\infty} \frac{x^{3}-3}{x-2} $$

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

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

Find the values of \(x\) such that $$ |2 \sqrt{x}-5|<0.01 $$

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

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

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{s \rightarrow 0} s\left(s^{2}-4\right) $$

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

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 \(e^{-x}=x^{2}\) has a solution in \((0,1)\).

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

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

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

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

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 \(2 ?\) (Hint: Find values of \(x\) such that \(|\sqrt{x}-2|<0.2 .)\) (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 \((0,1)\).

Show that $$ f(x)=\left\\{\begin{array}{cc} \frac{2 x^{2}+x-6}{x+2} & \text { if } x \neq-2 \\ -7 & \text { if } x=-2 \end{array}\right. $$ is continuous at \(x=-2\).

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

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{t \rightarrow \pi / 4} \sin (2 t) $$

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

Show that \(e^{-x}=x^{2}\) has a solution in \((0.5,1)\). Use the bisection method to find a solution that is accurate to two decimal places.

Let $$ f(x)=\left\\{\begin{array}{cc} \frac{x^{2}-9}{x-3} & \text { if } x \neq 3 \\ a & \text { if } x=3 \end{array}\right. $$ Which value must you assign to \(a\) so that \(f(x)\) is continuous at \(x=3 ?\)

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

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

Let $$ f(x)=\frac{e^{-x}}{2}, \quad x \geq 0. $$ (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).

Show that \(\cos x=x\) has a solution in \((0.5,1)\). Use the bisection method to find a solution that is accurate to two decimal places.

Let $$ f(x)=\left\\{\begin{array}{cc} \frac{3+2 x-x^{2}}{x-3} & \text { if } x \neq 3 \\ a & \text { if } x=3 \end{array}\right. $$ Which value must you assign to \(a\) so that \(f(x)\) is continuous at \(x=3\) ?

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

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

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

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

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

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

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