Chapter 13

A sequence \(a_{1}, a_{2}, a_{3}, \ldots\) has the limit \(L\) if the \(n\) th term \(a_{n}\) of the sequence can be made arbitrarily close to _____ by taking \(n\) to be sufficiently _____, If the limit exists, we say that the sequence __________; otherwise, the sequence __________.

Estimate the value of the limit by making a table of values. Check your work with a graph. $$\lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}$$

Suppose that $$\lim _{x \rightarrow a} f(x)=-3$$ $$\lim _{x \rightarrow a} g(x)=0$$ $$\lim _{x \rightarrow a} h(x)=8$$ 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) \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\) (b) \(\lim _{x \rightarrow a} \frac{2 f(x)}{h(x)-f(x)}\)

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

Estimate the value of the limit by making a table of values. Check your work with a graph. $$\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$$

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

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

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

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

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

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

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). $$\lim _{x \rightarrow-1} \frac{x-2}{x^{2}+4 x-3}$$

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

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

Complete the table of values (to five decimal places), and use the table to estimate the value of the limit. $$\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}$$ $$\begin{array}{|c|c|c|c||c|c|c|} \hline \boldsymbol{x} & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\ \hline \end{array}$$

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

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.

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

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

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

Complete the table of values (to five decimal places), and use the table to estimate the value of the limit. $$\lim _{x \rightarrow 0^{+}} x \ln x$$ $$\begin{array}{|c|c|c|c|c|c|} \hline x & 0.1 & 0.01 & 0.001 & 0.0001 & 0.00001 \\ \hline f(x) & & & & & \\ \hline \end{array}$$

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

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

(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.

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

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

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

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

(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. (b) Improve your estimates in part (a) by using eight rectangles.

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

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

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

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. $$y=3 x, \quad 0 \leq x \leq 5$$

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

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

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

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

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. $$y=2 x+1, \quad 1 \leq x \leq 3$$

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

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

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

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