Chapter 13

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

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\) $$f(x)=x^{2}$$

Determine each limit. $$\lim _{x \rightarrow-\infty} \frac{5 x+8 x^{2}}{3+2 x^{2}}$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 1}[\sqrt{x}(1+x)]$$

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=-2 x^{2}$$

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

Determine each limit. $$\lim _{x \rightarrow-\infty} \frac{-4 x^{3}+5 x^{2}+9}{5 x^{3}-2 x^{2}}$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 0}\left[2^{3 x}-\ln (x+1)\right]$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 3} f(x),\) where \(f(x)=\left\\{\begin{array}{ll}x+7 & \text { if } x \leq 3 \\ 5 x-5 & \text { if } x>3\end{array}\right.\)

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=3 x-1$$

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{1-7 x^{3}}{x^{2}+7 x^{3}}$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 3} \frac{\sqrt{x+1}}{\log _{2}(5 x+1)}$$

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=5-4 x$$

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

Determine each limit. $$\lim _{x \rightarrow-\infty} \frac{3 x^{4}-5 x+5}{x-2 x^{2}-x^{4}}$$

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=4 x-x^{2}$$

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

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{-4 x^{4}-x^{2}+8}{6 x^{4}-5 x}$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 1} f(x),\) where \(f(x)=\left\\{\begin{array}{ll}3 x-5 & \text { if } x \leq 1 \\ 6-2 x & \text { if } x>1\end{array}\right.\)

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=x^{2}-5 x$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 0} \sqrt[3]{x}$$

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{3 x^{3}+2 x-1}{2 x^{4}-3 x^{3}-2}$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 1} f(x),\) where \(f(x)=\left\\{\begin{array}{ll}e^{x} & \text { if } x \leq 1 \\ \sqrt{x} & \text { if } x>1\end{array}\right.\)

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=\sqrt{x}$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 0} \frac{\sin x-3 x}{x}$$

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{2 x^{2}-1}{3 x^{4}+2}$$

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

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=x^{3}$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 0} \frac{\sin x}{5 x}$$

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{2 x^{3}-x-3}{6 x^{2}-x-1}$$

Find the equation of the tangent line to the function \(f\) at the given point. Then graph the function and the tangent line together. $$f(x)=x^{2} \text { at }(-1,1)$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 0}(x \cot x)$$

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{x^{4}-x^{3}-3 x}{7 x^{2}+9}$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \frac{\sin x}{\sin 2 x}\)

Determine each limit, if it exists. $$\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{x^{2}}$$

Determine each limit. $$\lim _{x \rightarrow-\infty} \frac{-x^{3}-3 x+1}{4 x^{3}+5 x^{2}-x}$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \frac{\cos x-1}{x}\)

Determine each limit, if it exists. $$\lim _{x \rightarrow 0} \frac{\cos x-1}{3 x}$$

Find the equation of the tangent line to the function \(f\) at the given point. Then graph the function and the tangent line together. $$f(x)=x-x^{2} \text { at }(-1,-2)$$

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

Find the equation of the tangent line to the function \(f\) at the given point. Then graph the function and the tangent line together. $$f(x)=\frac{1}{2} x^{2}-2 \text { at }(2,0)$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 0} \frac{\cos x+2 \sin x-1}{3 x}$$

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

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \frac{e^{2 x}-1}{e^{x}-1}\)

Find the equation of the tangent line to the function \(f\) at the given point. Then graph the function and the tangent line together. $$f(x)=x^{3} \text { at }(1,1)$$

Evaluate each limit. (a) \(\lim _{x \rightarrow 1^{-}} \sqrt{1-x}\) (b) \(\lim _{x \rightarrow 1^{+}} \sqrt{1-x}\) (c) \(\lim _{x \rightarrow 1} \sqrt{1-x}\)

Evaluate each limit. (a) \(\lim _{x \rightarrow 4} \sqrt{x-3}\) (b) \(\lim _{x \rightarrow 2} \sqrt{x-3}\) (c) \(\lim _{x \rightarrow 3} \sqrt{x-3}\)

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

Find the equation of the tangent line to the function \(f\) at the given point. Then graph the function and the tangent line together. $$f(x)=\sqrt{x} \text { at }(4,2)$$

Evaluate each limit. (a) \(\lim _{x \rightarrow \infty} \sqrt[3]{x}\) (b) \(\lim _{x \rightarrow 0^{+}} \sqrt[3]{x}\) (c) \(\lim _{x \rightarrow 0} \sqrt[3]{x}\)