Chapter 13

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=\sqrt{2 x} ; x=2$$

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

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow 0^{+}} \frac{|x|}{x}$$

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=4-x^{2} ; x=-1$$

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

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow-3^{-}} \frac{|x+3|}{x+3}$$

Find the exact value of each integral, using formulas from geometry. Do not use a calculator. $$\int_{-3}^{3} \sqrt{9-x^{2}} d x$$

Find the slope of the tangent line to each curve when \(x\) has the given value. Do not use a calculator. $$f(x)=\frac{1}{x}+1 ; x=2$$

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

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow-\infty} \frac{6 x^{2}+1}{2 x^{2}+3}$$ (GRAPH CANNOT COPY).

Find the exact value of each integral, using formulas from geometry. Do not use a calculator. $$\int_{-4}^{0} \sqrt{16-x^{2}} d x$$

Find the equation of the tangent line to each curve when \(x\) has the given value. Verify your answer by graphing both \(f(x)\) and the tangent line with a calculator. $$f(x)=x^{2}+2 x ; x=3$$

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

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow \infty}\left(2+e^{-x}\right)$$ (GRAPH CANNOT COPY).

Find the exact value of each integral, using formulas from geometry. Do not use a calculator. $$\int_{1}^{3}(5-x) d x$$

Find the equation of the tangent line to each curve when \(x\) has the given value. Verify your answer by graphing both \(f(x)\) and the tangent line with a calculator. $$f(x)=6-x^{2} ; x=-1$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 1}\left(5 x^{8}-3 x^{2}+2\right)$$

Find the exact value of each integral, using formulas from geometry. Do not use a calculator. $$\int_{2}^{5}(1+2 x) d x$$

Find the equation of the tangent line to each curve when \(x\) has the given value. Verify your answer by graphing both \(f(x)\) and the tangent line with a calculator. $$f(x)=\frac{5}{x} ; x=2$$

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow \infty}\left(x+\frac{1}{x}\right)$$ (GRAPH CANNOT COPY).

Determine each limit, if it exists. $$\lim _{x \rightarrow 2}\left(x^{3}+4 x^{2}-5\right)$$

Use the table of values to predict \(\lim _{x \rightarrow 1} f(x)\) $$\begin{array}{|c|c|c|c|c|c|c|} \hline x & 0.9 & 0.99 & 0.999 & 1.001 & 1.01 & 1.1 \\ \hline f(x) & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \end{array}$$

Find the exact value of each integral, using formulas from geometry. Do not use a calculator. $$\int_{0}^{2} \sqrt{1-(x-1)^{2}} d x$$

Find the equation of the tangent line to each curve when \(x\) has the given value. Verify your answer by graphing both \(f(x)\) and the tangent line with a calculator. $$f(x)=\frac{-3}{x+1} ; x=1$$

Use a table and/or graph to find the asymptote\((s)\) of each function. $$f(x)=\frac{e^{x}}{e^{x}-1}$$

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

Use the table of values to predict \(\lim _{x \rightarrow 2} f(x)\) $$\begin{array}{|c|r|r|r|r|c|c|} \hline x & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & -1.3 & -1.05 & -1.002 & -0.997 & -0.993 & -0.985 \end{array}$$

Find the exact value of each integral, using formulas from geometry. Do not use a calculator. $$\int_{1}^{4}(2 x-1) d x$$

Find the equation of the tangent line to each curve when \(x\) has the given value. Verify your answer by graphing both \(f(x)\) and the tangent line with a calculator. $$f(x)=4 \sqrt{x} ; x=9$$

Use a table and/or graph to find the asymptote\((s)\) of each function. $$\lim _{x \rightarrow \infty}\left(x \sin \frac{1}{x^{2}}\right)$$

Determine each limit, if it exists. $$\lim _{x \rightarrow-1} \frac{2 x+3}{3 x+4}$$

Find the exact value of each integral, using formulas from geometry. Do not use a calculator. $$\int_{-3}^{2}|x+1| d x$$

Find the equation of the tangent line to each curve when \(x\) has the given value. Verify your answer by graphing both \(f(x)\) and the tangent line with a calculator. $$f(x)=\sqrt{x} ; x=25$$

Use a table and/or graph to find the asymptote\((s)\) of each function. $$\lim _{x \rightarrow \infty}(\sqrt{x^{2}+x}-x)$$

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

Find the exact value of each integral, using formulas from geometry. Do not use a calculator. $$\int_{-2}^{4}|x-2| d x$$

By considering the graph of the function but not calculating any limits, give the value of \(f^{\prime}(2)\) for each function. $$f(x)=5$$

Use a table and/or graph to find the asymptote\((s)\) of each function. $$\lim _{x \rightarrow \infty}(x-\sqrt{x^{2}+5})$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3}$$

By considering the graph of the function but not calculating any limits, give the value of \(f^{\prime}(2)\) for each function. $$f(x)=x$$

Use a table and/or graph to find the asymptote\((s)\) of each function. $$f(x)=\frac{e^{x}}{e^{x}-1}$$

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

Complete each table and use the results to predict the indicated limit, if it exists. $$\text { If } f(x)=\frac{\sqrt{x}-3}{x-3}, \text { find } \lim _{x \rightarrow 3} f(x)$$ $$\begin{array}{|c|l|l|l|l|l|l|} \hline x & 2.9 & 2.99 & 2.999 & 3.001 & 3.01 & 3.1 \\ \hline f(x) & & & & & & \end{array}$$

By considering the graph of the function but not calculating any limits, give the value of \(f^{\prime}(2)\) for each function. $$f(x)=-x$$

Determine each limit, if it exists. $$\lim _{x \rightarrow-2} \frac{x^{2}-x-6}{x+2}$$

Complete each table and use the results to predict the indicated limit, if it exists. If \(f(x)=\frac{\sqrt{x}-2}{x-1},\) find \(\lim _{x \rightarrow 1} f(x)\) $$\begin{array}{|c|c|c|c|c|c|c|} \hline x & 0.9 & 0.99 & 0.999 & 1.001 & 1.01 & 1.1 \\ \hline f(x) & & & & & & \end{array}$$

By considering the graph of the function but not calculating any limits, give the value of \(f^{\prime}(2)\) for each function. $$f(x)=3 x+4$$

Use a table and/or graph to find the asymptote\((s)\) of each function. $$f(x)=\frac{x-\cos x}{x+\sin x}$$

Determine each limit, if it exists. $$\lim _{x \rightarrow 5} \frac{x^{2}-3 x-10}{x-5}$$

Complete each table and use the results to predict the indicated limit, if it exists. $$\text { If } f(x)=\frac{x^{3}+3 x^{2}+x+3}{x+3}, \text { find } \lim _{x \rightarrow-3} f(x)$$ $$\begin{array}{|c|c|c|c|c|c|c|} \hline x & -3.1 & -3.01 & -3.001 & -2.999 & -2.99 & -2.9 \\ \hline f(x) & & & & & & \end{array}$$