Chapter 13

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 2^{-}} \frac{x+1}{x^{2}-x-2}$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow-\infty} \frac{(x-3)(x+2)}{2 x^{2}+x+1}$$

Determine whether or not the function is continuous at the given number. $$f(x)=\left\\{\begin{array}{ll} x^{2}-x & \text { if } x \leq 0 \\ 2 x^{2} & \text { if } x>0 \end{array} \text { at } x=0\right.$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 1} \sqrt{x^{3}+6 x^{2}+2 x+5}$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty} \frac{(2 x+1)(3 x-2)}{3 x^{2}+2 x-5}$$

Determine whether or not the function is continuous at the given number. $$g(x)=\left\\{\begin{array}{ll} x^{3}-x+1 & \text { if } x<2 \\ 3 x^{2}-2 x-1 & \text { if } x \geq 2 \end{array} \text { at } x=2\right.$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 2} \sqrt{x^{2}+x+3}$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty}\left(3 x-\frac{1}{x^{2}}\right)$$

Determine whether or not the function is continuous at the given number. $$f(x)=|x-3| \text { at } x=3$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 1^{+}}(\sqrt{x-1}+3)$$

Determine whether or not the function is continuous at the given number. $$k(x)=-|x+2|+3 \text { at } x=-2$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 3^{-}} \sqrt{-3-x}$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow-\infty}\left(\frac{3 x}{x+2}+\frac{2 x}{x-1}\right)$$

Determine all numbers at which the function is continuous. $$f(x)=\left\\{\begin{array}{ll} \frac{x^{2}+x-2}{x^{2}-4 x+3} & \text { if } x \neq 1 \\ -3 / 2 & \text { if } x=1 \end{array}\right.$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-25^{+}}(\sqrt{5+2 x}+x)$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty}\left(\frac{x}{x^{2}+1}+\frac{2 x^{2}}{x^{3}+x}\right)$$

Determine all numbers at which the function is continuous. $$g(x)=\left\\{\begin{array}{ll} \frac{x^{2}-x-6}{x^{2}-4} & \text { if } x \neq-2 \\ 5 / 4 & \text { if } x=-2 \end{array}\right.$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty} \frac{2 x}{\sqrt{x^{2}-2 x}}$$

Determine all numbers at which the function is continuous. $$f(x)=\left\\{\begin{array}{ll} x^{2}+1 & \text { if } x<0 \\ x & \text { if } 02 \end{array}\right.$$

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow 3} \frac{5 x-4}{2 x-1}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 3} \frac{\sqrt{x}-\sqrt{3}}{x-3}$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow-\infty} \frac{x}{\sqrt{x^{2}-1}}$$

Determine all numbers at which the function is continuous. $$h(x)=\left\\{\begin{array}{ll} 1 / x & \text { if } x<1 \text { and } x \neq 0 \\ x^{2} & \text { if } x \geq 1 \end{array}\right.$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 25} \frac{\sqrt{x}-5}{x-25}$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow-\infty} \frac{3 x-2}{\sqrt{2 x^{2}+1}}$$

Justify your answers. Taxis in New York City cost \(\$ 2\) plus 30 cents for each \(1 / 5\) of a mile (or portion thereof). Let \(f(x)\) be the cost of traveling \(x\) miles. Is \(f\) continuous on the interval [0,3]\(?\)

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow 2} \frac{x^{2}-5 x+6}{x^{2}-6 x+8}$$

Find the limit if it exists. If the limit does not exist, explain why. \(\lim _{x \rightarrow 0}\left(\frac{1 /(x+5)-1 / 5}{x}\right) \quad[\text {Hint}:\) Write the expression in parentheses as a single fraction.]

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty} \frac{3 x-2}{\sqrt{2 x^{2}+1}}$$

Justify your answers. On a four and a half hour flight from Chicago to Seattle, let \(h(x)\) be the height of the plane above the ground at time \(x\) hours. Is \(h\) continuous on the interval [0,4.5]\(?\)

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow-2} \frac{x^{2}+3 x+2}{x^{2}-x-6}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0}\left(\frac{2 /(x+6)-1 / 3}{x}\right)$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty} \frac{\sqrt{2 x^{2}+1}}{3 x-5}$$

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow 4} \frac{x-6}{x-4}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 1}\left[\frac{1}{x-1}-\frac{2}{x^{2}-1}\right]$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow-\infty} \frac{\sqrt{2 x^{2}+1}}{3 x-5}$$

Justify your answers. Postage on a letter from the United States to Germany is 80 cents for each ounce (or fraction thereof) for letters weighing up to 8 ounces. Let \(f(x)\) be the postage for a letter weighing \(x\) ounces. At what points on the interval (0,8] is \(f\) continuous?

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow-3} \frac{(x-3)(x+3)(x+4)}{(x-4)(x+3)(x+8)}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-2}\left[\frac{x^{2}}{x+2}+\frac{2 x}{x+2}\right]$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow-\infty} \frac{\sqrt{3 x^{2}+3}}{x+3}$$

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow 2^{+}} \sqrt{x^{2}-4}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0} \frac{x^{2}}{|x|}$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty} \frac{\sqrt{3 x^{2}+2 x}}{2 x+1}$$

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow 3^{-}} \sqrt{9-x^{2}}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-2}|x+2|$$

Show that the function f(x)=\frac{x^{4}-5 x^{2}+4}{x-1} is not continuous on [-3,3] but does satisfy the conclusion of the Intermediate Value Theorem (that is, if \(k\) is a number between \(f(-3)\) and \(f(3),\) there is a number \(c\) between -3 and 3 such that \(f(c)=k\) ). [ Hint: What can be said about \(f\) on the intervals \([-3,-2] \text { and }[2,3] ?]\)

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty} \frac{x^{2}+2 x+1}{\sqrt{x^{4}+2 x}}$$

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow 4} \frac{-6}{(x-4)^{2}}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-3^{+}}\left(\frac{|x+3|}{x+3}+\sqrt{x+3}+1\right)$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty} \frac{\sqrt{x^{6}-x^{2}}}{2 x^{3}}$$