Chapter 13

Use a calculator to estimate \(f^{\prime}(a)\) for the given value of \(a\). $$f(x)=e^{x} ; a=0$$

Use a table and/or graph to find the asymptote\((s)\) of each function. $$f(x)=5-e^{-x}$$

Complete each table and use the results to predict the indicated limit, if it exists. If \(f(x)=\frac{\sin 2 x}{x},\) find \(\lim _{x \rightarrow 0} f(x)\) $$\begin{array}{|c|c|c|c|c|c|c|} \hline x & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\ \hline f(x) & & & & & & \end{array}$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 1} \frac{x^{2}+x-2}{x-1}$$

Use a calculator to estimate \(f^{\prime}(a)\) for the given value of \(a\). $$f(x)=\sin x ; a=0$$

Determine each limit. \(f(x)=\left\\{\begin{array}{ll}2 x+3 & \text { if } x<1 \\ 4 & \text { if } x=1 \\ x^{2} & \text { if } x>1\end{array}\right.\) (a) \(\lim _{x \rightarrow 1^{+}} f(x)\) (b) \(\lim _{x \rightarrow 1^{-}} f(x)\)

Complete each table and use the results to predict the indicated limit, if it exists. $$\text { If } f(x)=\frac{\sin 5 x}{2 x}, \text { find } \lim _{x \rightarrow 0} f(x)$$ $$\begin{array}{|c|c|c|c|c|c|c|} \hline x & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\ \hline f(x) & & & & & & \end{array}$$

Use a calculator to estimate \(f^{\prime}(a)\) for the given value of \(a\). $$f(x)=\frac{10 x}{1+0.25 x^{2}} ; a=2$$

Determine each limit. \(f(x)=\left\\{\begin{array}{ll}7 x & \text { if } x \leq 2 \\ x-1 & \text { if } x>2\end{array}\right.\) (a) \(\lim _{x \rightarrow 2^{+}} f(x)\) (b) \(\lim _{x \rightarrow 2^{-}} f(x)\)

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 5}|2 x-4|\)

Determine each limit, if it exists. $$\lim _{x \rightarrow 1} \frac{(x-1)^{2}}{x^{2}+x-2}$$

Use a calculator to estimate \(f^{\prime}(a)\) for the given value of \(a\). $$f(x)=\frac{1}{1+x^{2}} ; a=0$$

Determine each limit. \(f(x)=\frac{1}{(1+x)^{3}}\) (a) \(\lim _{x \rightarrow-1^{+}} f(x)\) (b) \(\lim _{x \rightarrow-1^{-}} f(x)\)

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow-1} \sqrt{5+3 x}\)

Determine each limit, if it exists. $$\lim _{x \rightarrow-1} \frac{(x+1)^{2}}{2 x^{2}-x-3}$$

Use a calculator to estimate \(f^{\prime}(a)\) for the given value of \(a\). $$f(x)=x \cos x ; a=\frac{\pi}{4}$$

Determine each limit. \(f(x)=\frac{x}{(4-x)^{3}}\) (a) \(\lim _{x \rightarrow 4^{+}} f(x)\) (b) \(\lim _{x \rightarrow 4^{-}} f(x)\)

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 5} \frac{x^{2}-3 x-10}{x-5}\)

Determine each limit, if it exists. $$\lim _{x \rightarrow 2} \frac{x^{3}-8}{x^{4}-16}$$

Use a calculator to estimate \(f^{\prime}(a)\) for the given value of \(a\). $$f(x)=x e^{x} ; a=1$$

Determine each limit. \(f(x)=\frac{1}{(x-3)^{2}}\) (a) \(\lim _{x \rightarrow 3^{+}} f(x)\) (b) \(\lim _{x \rightarrow 3^{-}} f(x)\)

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow-4} \frac{x^{2}+5 x+4}{x+4}\)

Determine each limit, if it exists. $$\lim _{x \rightarrow 1} \frac{x^{3}-1}{x^{4}-1}$$

Determine each limit. \(f(x)=\frac{x}{(x+3)^{3}}\) (a) \(\lim _{x \rightarrow-3^{+}} f(x)\) (b) \(\lim _{x \rightarrow-3^{-}} f(x)\)

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow-2} \frac{x^{2}+2}{x+2}\)

Determine each limit, if it exists. $$\lim _{x \rightarrow 4} \frac{x-4}{\sqrt{x}-2}$$

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{3 x}{5 x-1}$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 2} \frac{\frac{1}{x}-\frac{1}{2}}{x-2}$$

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{5 x}{3 x-1}$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 2} \frac{x^{2}-x-2}{x-2}\)

Determine each limit, if it exists. $$\lim _{x \rightarrow 3} 2^{\left(x^{2}-7\right)}$$

Determine each limit. $$\lim _{x \rightarrow-\infty} \frac{2 x+3}{4 x-7}$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 2} \frac{x^{2}-3 x+2}{x-2}\)

Determine each limit, if it exists. $$\lim _{x \rightarrow 4} 5^{\left(x^{2}-13\right)}$$

The position in feet of a car along a straight racetrack after \(t\) seconds is approximated by \(s(t)\) Find the car's velocity in feet per second after 3 seconds. $$s(t)=9 t^{2}$$

Determine each limit. $$\lim _{x \rightarrow-\infty} \frac{8 x+2}{2 x-5}$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow-3} \frac{2 x^{2}+5 x-3}{x+3}\)

Determine each limit, if it exists. $$\lim _{x \rightarrow 1} 9^{1 /(x+1)}$$

The position in feet of a car along a straight racetrack after \(t\) seconds is approximated by \(s(t)\) Find the car's velocity in feet per second after 3 seconds. $$s(t)=50 \sqrt{t}$$

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{x^{2}+2 x}{2 x^{3}-2 x+1}$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow-2} \frac{x+2}{x^{2}-4}\)

Determine each limit, if it exists. $$\lim _{x \rightarrow 3} 5^{\sqrt{x+1}}$$

The position in feet of a car along a straight racetrack after \(t\) seconds is approximated by \(s(t)\) Find the car's velocity in feet per second after 3 seconds. $$s(t)=3 t^{3}-t^{2}$$

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{x^{2}+2 x-5}{3 x^{2}+2}$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 1} \frac{x-1}{x^{2}-1}\)

Determine each limit, if it exists. $$\lim _{x \rightarrow 5}\left[\log _{3}(2 x-1)\right]$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 3} \frac{x-3}{x^{2}-6 x+9}\)

The position in feet of a car along a straight racetrack after \(t\) seconds is approximated by \(s(t)\) Find the car's velocity in feet per second after 3 seconds. $$s(t)=4 t^{2}+5 t+1$$

Determine each limit. $$\lim _{x \rightarrow-\infty} \frac{1-2 x+3 x^{2}}{2 x^{2}+5 x}$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 4}\left[\log _{2}(14+\sqrt{x})\right]$$