Chapter 13

Evaluate the limit if it exists. $$\lim _{t \rightarrow-3} \frac{t^{2}-9}{2 t^{2}+7 t+3}$$

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

Find the derivative of the function at the given number. $$f(x)=1-3 x^{2} \text { at } 2$$

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

Find the area of the region that lies under the graph of \(f\) over the given interval. $$f(x)=x+x^{2}, \quad 0 \leq x \leq 1$$

Evaluate the limit if it exists. $$\lim _{h \rightarrow 0} \frac{\sqrt{1+h}-1}{h}$$

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

Find the derivative of the function at the given number. $$f(x)=2-3 x+x^{2} \quad \text { at }-1$$

Find the area of the region that lies under the graph of \(f\) over the given interval. $$f(x)=x^{3}+2, \quad 0 \leq x \leq 5$$

Evaluate the limit if it exists. $$\lim _{h \rightarrow 0} \frac{(2+h)^{3}-8}{h}$$

Find the derivative of the function at the given number. $$g(x)=x^{4} \quad \text { at } 1$$

Find the area of the region that lies under the graph of \(f\) over the given interval. $$f(x)=4 x^{3}, \quad 2 \leq x \leq 5$$

Evaluate the limit if it exists. $$\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}$$

Find the derivative of the function at the given number. $$g(x)=2 x^{2}+x^{3} \quad \text { at } 1$$

Find the area of the region that lies under the graph of \(f\) over the given interval. $$f(x)=x+6 x^{2}, \quad 1 \leq x \leq 4$$

Evaluate the limit if it exists. $$\lim _{x \rightarrow 7} \frac{\sqrt{x+2}-3}{x-7}$$

Use a table of values to estimate the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+4 x}}{4 x+1}$$

Find the derivative of the function at the given number. $$F(x)=\frac{1}{\sqrt{x}} \quad \text { at } 4$$

Find the area of the region that lies under the graph of \(f\) over the given interval. $$f(x)=20-2 x^{2}, \quad 2 \leq x \leq 3$$

Evaluate the limit if it exists. $$\lim _{h \rightarrow 0} \frac{(3+h)^{-1}-3^{-1}}{h}$$

Use a table of values to estimate the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow \infty}(\sqrt{9 x^{2}+x}-3 x)$$

Find the derivative of the function at the given number. $$G(x)=1+2 \sqrt{x} \text { at } 4$$

Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 1} \frac{x^{3}+x^{2}+3 x-5}{2 x^{2}-5 x+3}$$

When we approximate areas using rectangles as in Example \(1,\) then the more rectangles we use, the more accurate the answer. The following TI- 83 program finds the approximate area under the graph of \(f\) on the interval \([a, b]\) using \(n\) rectangles. To use the program, first store the function \(f\) in \(Y_{1}\). The program prompts you to enter \(\mathrm{N}\), the number of rectangles, and \(\mathrm{A}\) and \(\mathrm{B}\), the endpoints of the interval. (a) Approximate the area under the graph of \(f(x)=x^{5}+2 x+3\) on \([1,3],\) using \(10,20\) and 100 rectangles. (b) Approximate the area under the graph of \(f\) on the given interval, using 100 rectangles. (i) \(f(x)=\sin x, \quad\) on \([0, \pi]\) (ii) \(f(x)=e^{-x^{2}}, \quad\) on \([-1,1]\) (GRAPH CANT COPY)

Evaluate the limit if it exists. $$\lim _{x \rightarrow-4} \frac{\frac{1}{4}+\frac{1}{x}}{4+x}$$

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

Find \(f^{\prime}(a),\) where \(a\) is in the domain of \(f .\) $$f(x)=x^{2}+2 x$$

Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 0} \frac{x^{2}}{\cos 5 x-\cos 4 x}$$

Evaluate the limit if it exists. $$\lim _{t \rightarrow 0}\left(\frac{1}{t}-\frac{1}{t^{2}+t}\right)$$

Use a table of values to estimate the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{3 x}$$

Find \(f^{\prime}(a),\) where \(a\) is in the domain of \(f .\) $$f(x)=-\frac{1}{x^{2}}$$

Find the limit and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1} \frac{x^{2}-1}{\sqrt{x}-1}$$

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{1+n}{n+n^{2}}$$

Find \(f^{\prime}(a),\) where \(a\) is in the domain of \(f .\) $$f(x)=\frac{x}{x+1}$$

Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 2} \frac{x^{3}+6 x^{2}-5 x+1}{x^{3}-x^{2}-8 x+12}$$

Find the limit and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{(4+x)^{3}-64}{x}$$

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{5 n}{n+5}$$

Find \(f^{\prime}(a),\) where \(a\) is in the domain of \(f .\) $$f(x)=\sqrt{x-2}$$

Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 0} \cos \frac{1}{x}$$

Find the limit and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow-1} \frac{x^{2}-x-2}{x^{3}-x}$$

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{n^{2}}{n+1}$$

(a) If \(f(x)=x^{3}-2 x+4,\) find \(f^{\prime}(a)\) (b) Find equations of the tangent lines to the graph of \(f\) at the points whose \(x\) -coordinates are \(0,1,\) and 2 (c) Graph \(f\) and the three tangent lines.

Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 0} \sin \frac{2}{x}$$

Find the limit and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1} \frac{x^{8}-1}{x^{5}-x}$$

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{n-1}{n^{3}+1}$$

(a) If \(g(x)=1 /(2 x-1),\) find \(g^{\prime}(a)\) (b) Find equations of the tangent lines to the graph of \(g\) at the points whose \(x\) -coordinates are \(-1,0,\) and 1 (c) Graph \(g\) and the three tangent lines.

(a) Estimate the value of $$\lim _{x \rightarrow 0} \frac{x}{\sqrt{1+3 x}-1}$$ by graphing the function \(f(x)=x /(\sqrt{1+3 x}-1)\) (b) Make a table of values of \(f(x)\) for \(x\) close to \(0,\) and guess the value of the limit. (c) Use the Limit Laws to prove that your guess is correct.

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{1}{3^{n}}$$

If a ball is thrown straight up with a velocity of \(40 \mathrm{ft} / \mathrm{s}\), its height (in feet) after \(t\) seconds is given by \(y=40 t-16 t^{2} .\) Find the instantaneous velocity when \(t=2\)

Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 0} \frac{1}{1+e^{1 / x}}$$