Chapter 13

Let \(f\) be a function defined on some interval \((a, \infty) .\) Then $$\lim _{x \rightarrow \infty} f(x)=L$$ means that the values of \(f(x)\) can be made arbitrarily close to ______ by taking ______ sufficiently large. In this case the line \(y=L\) is called a ______ _______ of the function \(y=f(x) .\) For example, \(\lim _{x \rightarrow \infty} \frac{1}{x}=\) ________, and the line \(y=\) _______ is a horizontal asymptote.

Suppose the following limits exist: $$\lim _{x \rightarrow a} f(x) \quad \text { and } \quad \lim _{x \rightarrow a} g(x)$$ Then \(\lim [f(x)+g(x)]=\text{____}+\text{____},\) and \(\lim _{x \rightarrow a}[f(x) g(x)]=\text{____}, \text{____}\) These formulas can be stated verbally as follows: The limit of a sum is the _____ of the limits, and the limit of a product is the _______ of the limits.

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 ________.

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

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

Estimating Limits Numerically and Graphically 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. $$f(x)=5-2 x, \quad \text { at }(-3,11)$$

Using Limit Laws 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) \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\) (h) \(\lim _{x \rightarrow a} \frac{2 f(x)}{h(x)-f(x)}\)

Estimating Limits Numerically 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. $$f(x)=4 x^{2}-3 x, \quad \text { at }(-1,7)$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 5} x$$

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

Estimating Limits Numerically Complete the table of values (to five decimal places), and use the table to estimate the value of the limit. $$\lim _{x \rightarrow 2} \frac{x-2}{x^{2}+x-6}$$ $$\begin{array}{|c|c|c|c||c|c|c|} \hline x & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & & & & & \\ \hline \end{array}$$

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

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 0} 3$$

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

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

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

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{t \rightarrow 3} 4 t$$

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. $$f(x)=x^{3}+1, \quad \text { at }(2,9)$$

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

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. $$f(x)=\frac{5}{x+2}, \quad \text { at }(3,1)$$

In these exercises we estimate the area under the graph of a function by using rectangles. (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). $$\lim _{x \rightarrow 4}\left(5 x^{2}-2 x+3\right)$$

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

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

In these exercises we estimate the area under the graph of a function by using rectangles. (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.

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 0}\left(3 x^{3}-2 x^{2}+5\right)$$

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

Estimating Limits Numerically and Graphically 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. $$f(x)=-2 x^{2}+1, \quad \text { at }(2,-7)$$

In these exercises we estimate the area under the graph of a function by using rectangles. (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 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. $$\lim _{t \rightarrow \infty} \frac{8 t^{3}+t}{(2 t-1)\left(2 t^{2}+1\right)}$$

Estimating Limits Numerically and Graphically 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. $$f(x)=4 x^{2}-3, \quad \text { at }(-1,1)$$

In these exercises we estimate the area under the graph of a function by using rectangles. (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 and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 2} \frac{2-x}{x^{2}+1}$$

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

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

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

13-14 - Finding the Area Under A Curve 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 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. $$\lim _{x \rightarrow \infty} \frac{x^{4}}{1-x^{2}+x^{3}}$$

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

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

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