Chapter 12

Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=3 x+4 \quad \text { at }(1,7)$$

Suppose that \(\lim _{x \rightarrow a} f(x)=-3 \quad \lim _{x \rightarrow a} g(x)=0 \quad \lim _{x \rightarrow a} h(x)=8\) Find the value of the given limit. If the limit does not exist, explain why. (a) \(\lim _{x \rightarrow a}[f(x)+h(x)]\) (b) \(\lim _{x \rightarrow a}[f(x)]^{2}\) (c) \(\lim _{x \rightarrow a} \sqrt[3]{h(x)}\) (d) \(\lim _{x \rightarrow a} \frac{1}{f(x)}\) (e) \(\lim _{x \rightarrow a} \frac{f(x)}{h(x)}\) (f) \(\lim _{x \rightarrow a} \frac{g(x)}{f(x)}\) (g) \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\) (h) \(\lim _{x \rightarrow a} \frac{2 f(x)}{h(x)-f(x)}\)

Complete the table of values (to five decimal places) and use the table to estimate the value of the limit. $$\lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4}$$ $$\begin{array}{|c|c|c|c||c|c|c|} \hline x & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \\ \hline f(x) & & & & & & \\ \hline \end{array}$$

Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=5-2 x \text { at }(-3,11)$$

Find the limit. $$\lim _{x \rightarrow \infty} \frac{6}{x}$$

Complete the table of values (to five decimal places) and use the table to estimate the value of the limit. $$\lim _{x \rightarrow 1} \frac{x-1}{x^{3}-1}$$ $$\begin{array}{|c|c|c|c||c|c|c|} \hline x & 0.9 & 0.99 & 0.999 & 1.001 & 1.01 & 1.1 \\ \hline f(x) & & & & & & \\ \hline \end{array}$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 4}\left(5 x^{2}-2 x+3\right)$$

Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=4 x^{2}-3 x \quad \text { at }(-1,7)$$

Find the limit. $$\lim _{x \rightarrow \infty} \frac{3}{x^{4}}$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 3}\left(x^{3}+2\right)\left(x^{2}-5 x\right)$$

Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=1+2 x-3 x^{2} \quad \text { at }(1,0)$$

Find the limit. $$\lim _{x \rightarrow \infty} \frac{2 x+1}{5 x-1}$$

Complete the table of values (to five decimal places) and use the table to estimate the value of the limit. $$\lim _{x \rightarrow 0} \frac{\sin x}{x}$$ $$\begin{array}{|c|c|c||c|c|c|} \hline x & \pm 1 & \pm 0.5 & \pm 0.1 & \pm 0.05 & \pm 0.01 \\ \hline f(x) & & & & & \\ \hline \end{array}$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow-1} \frac{x-2}{x^{2}+4 x-3}$$

Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=2 x^{3} \quad \text { at }(2,16)$$

Find the limit. $$\lim _{x \rightarrow \infty} \frac{2-3 x}{4 x+5}$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 1}\left(\frac{x^{4}+x^{2}-6}{x^{4}+2 x+3}\right)^{2}$$

Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=\frac{6}{x+1} \quad \text { at }(2,2)$$

(a) Estimate the area under the graph of \(f(x)=1 / x\) from \(x=1\) to \(x=5\) using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.

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

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow-4} \frac{x+4}{x^{2}+7 x+12}$$

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. $$y=x+x^{2} \quad \text { at }(-1,0)$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{t \rightarrow-2}(t+1)^{9}\left(t^{2}-1\right)$$

(a) Estimate the area under the graph of \(f(x)=25-x^{2}\) from \(x=0\) to \(x=5\) using five approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.

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

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1} \frac{x^{3}-1}{x^{2}-1}$$

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. $$y=2 x-x^{3} \quad \text { at }(1,1)$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{u \rightarrow-2} \sqrt{u^{4}+3 u+6}$$

(a) Estimate the area under the graph of \(f(x)=1+x^{2}\) from \(x=-1\) to \(x=2\) using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles. (b) Repeat part (a) using left endpoints.

Find the limit. $$\lim _{t \rightarrow \infty} \frac{8 t^{3}+t}{(2 t-1)\left(2 t^{2}+1\right)}$$

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{5^{x}-3^{x}}{x}$$

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

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. $$y=\frac{x}{x-1} \quad \text { at }(2,2)$$

(a) Estimate the area under the graph of \(f(x)=e^{-x}\) \(0 \leq x \leq 4,\) using four approximating rectangles and taking the sample points to be (i) right endpoints (ii) left endpoints In each case, sketch the curve and the rectangles. In each case, sketch the curve and the rectangles. (b) Improve your estimates in part (a) by using eight rectangles.

Find the limit. $$\lim _{r \rightarrow \infty} \frac{4 r^{3}-r^{2}}{(r+1)^{3}}$$

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{\sqrt{x+9}-3}{x}$$

Evaluate the limit, if it exists. $$\lim _{x \rightarrow-4} \frac{x^{2}+5 x+4}{x^{2}+3 x-4}$$

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. $$y=\frac{1}{x^{2}} \quad \text { at }(-1,1)$$

Use the definition of area as a limit to find the area of the region that lies under the curve. Check your answer by sketching the region and using geometry. $$y=3 x, \quad 0 \leq x \leq 5$$

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

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1}\left(\frac{1}{\ln x}-\frac{1}{x-1}\right)$$

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. $$y=\sqrt{x+3} \text { at }(1,2)$$

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

Use the definition of area as a limit to find the area of the region that lies under the curve. Check your answer by sketching the region and using geometry. $$y=2 x+1, \quad 1 \leq x \leq 3$$

Find the limit. $$\lim _{t \rightarrow \infty}\left(\frac{1}{t}-\frac{2 t}{t-1}\right)$$

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{\tan 2 x}{\tan 3 x}$$

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

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. $$y=\sqrt{1+2 x} \text { at }(4,3)$$

Find the area of the region that lies under the graph of \(f\) over the given interval. $$f(x)=3 x^{2}, \quad 0 \leq x \leq 2$$

Find the limit. $$\lim _{x \rightarrow-\infty}\left(\frac{x-1}{x+1}+6\right)$$