Chapter 13

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 \(f(c) ?\) $$\lim _{x \rightarrow-2} \frac{2 x}{(x+2)^{3}}$$

For what values of \(b\) is the function $$f(x)=\left\\{\begin{array}{ll} b x+4 & \text { if } x \leq 3 \\ b x^{2}-2 & \text { if } x>3 \end{array}\right.$$ continuous at \(x=3 ?\)

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty} \frac{1-\sqrt{x}}{1+\sqrt{x}} \quad[\text {Hint: Rationalize the denominator.}]$$

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 \(f(c) ?\) $$\lim _{x \rightarrow 0} \frac{x}{e^{x}-1}$$

Show that \(f(x)=\sqrt{|x|}\) is continuous at \(x=0\)

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

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 \(f(c) ?\) $$\lim _{x \rightarrow 0} \frac{e^{2 x}+e^{x}-2}{e^{x}-1}$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0}\left[\frac{1}{x \sqrt{x+1}}-\frac{1}{x}\right]$$

A function \(f\) that is not defined at \(x=c\) is said to have \(a\) removable discontinuity at \(x=c\) if there is a function \(g\) such that \(g(c)\) is defined, \(g\) is continuous at \(x=c,\) and \(g(x)=f(x)\) for \(x \neq c .\) In Exercises \(39-43,\) show that the function \(f\) has a removable discontinuity by finding an appropriate function g. $$f(x)=\frac{x-1}{x^{2}-1}$$

Use the Infinite Limit Theorem and the properties of limits to find the limit. $$\lim _{x \rightarrow \infty}(\sqrt{x^{2}+1}-x)\left[\text { Hint: Multiply by } \frac{\sqrt{x^{2}+1}+x}{\sqrt{x^{2}+1}+x}\right]$$

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0}\left[\frac{|x|}{x}-\frac{x}{|x|}\right]$$

A function \(f\) that is not defined at \(x=c\) is said to have \(a\) removable discontinuity at \(x=c\) if there is a function \(g\) such that \(g(c)\) is defined, \(g\) is continuous at \(x=c,\) and \(g(x)=f(x)\) for \(x \neq c .\) In Exercises \(39-43,\) show that the function \(f\) has a removable discontinuity by finding an appropriate function g. $$f(x)=\frac{x^{2}}{|x|}$$

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 \(f(c) ?\) $$\lim _{x \rightarrow 5} \frac{\ln x-\ln 5}{x-5}$$

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

A function \(f\) that is not defined at \(x=c\) is said to have \(a\) removable discontinuity at \(x=c\) if there is a function \(g\) such that \(g(c)\) is defined, \(g\) is continuous at \(x=c,\) and \(g(x)=f(x)\) for \(x \neq c .\) In Exercises \(39-43,\) show that the function \(f\) has a removable discontinuity by finding an appropriate function g. $$f(x)=\frac{2-\sqrt{x}}{4-x}$$

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 \(f(c) ?\) $$\lim _{x \rightarrow 0}(x \ln |x|)$$

Find $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=x^{2}$$

A function \(f\) that is not defined at \(x=c\) is said to have \(a\) removable discontinuity at \(x=c\) if there is a function \(g\) such that \(g(c)\) is defined, \(g\) is continuous at \(x=c,\) and \(g(x)=f(x)\) for \(x \neq c .\) In Exercises \(39-43,\) show that the function \(f\) has a removable discontinuity by finding an appropriate function g. $$f(x)=\frac{\sin x}{x} \quad[\text {Hint}: \text { See Example 4 on pages } 882-883 .]$$

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 \(f(c) ?\) $$\lim _{x \rightarrow \pi / 2} x \sin x$$

Find $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=x^{3}$$

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

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 \(f(c) ?\) $$\lim _{x \rightarrow \pi / 2} x \cos x$$

Find $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=x^{2}+x+1$$

A ranger leaves his truck at a parking lot at the trail head at 8: 00 A.M. and hikes 11 miles to a fire tower, arriving there at noon. He stays overnight and starts back along the same trail at 8: 00 A.M., arriving at the parking lot at noon. Show that there is a point on the trail that he passes at exactly the same time on both days. [Hint: Let \(f(t)\) be his distance from the parking lot at time \(t\) on the first day, and let \(g(t)\) be his distance from the parking lot at time \(t\) on the second day. Use an appropriate theorem to solve the equation \(f(t)=g(t) .]\)

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

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 \(f(c) ?\) $$\lim _{x \rightarrow 1} \tan \pi x$$

Find $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=\sqrt{x}$$

Let \([x]\) denote the greatest integer function (see Example 7 on page 145 ) and find: (a) \(\lim _{x \rightarrow \infty} \frac{|x|}{x}\) (b) \(\lim _{x \rightarrow-\infty} \frac{|| x||}{x}\)

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 \(f(c) ?\) $$\lim _{x \rightarrow 1^{-}} \frac{|x-1|}{x-1}$$

Find \(\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h},\) if it exists. $$f(x)=|x|$$

Use the change of base formula for logarithms (Special Topics \(5.4 . \mathrm{A}\) ) to show that \(\lim _{x \rightarrow \infty} \frac{\ln x}{\log x}=\ln 10\)

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 \(f(c) ?\) $$\lim _{x \rightarrow-5} \frac{|x+5|}{x+5}$$

Find \(\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h},\) if it exists. $$f(x)=x|x|$$

Find \(\lim _{x \rightarrow \infty} \frac{\sqrt{x+\sqrt{x+\sqrt{x}}}}{\sqrt{x+1}}\)

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 \(f(c) ?\) $$\lim _{x \rightarrow 0} \cos \left(\frac{\pi}{x}\right)$$

Find the rule of the derivative of the function \(f .\) [See Example 7 and the remarks following it.] $$f(x)=2 x+3$$

Let \(f(x)\) be a nonzero polynomial with leading coefficient \(a\) and let \(g(x)\) be a nonzero polynomial with leading coefficient \(c .\) Prove that (a) If \(\operatorname{deg} f(x)<\operatorname{deg} g(x),\) then \(\quad \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=0\) (b) If \(\operatorname{deg} f(x)=\operatorname{deg} g(x),\) then \(\quad \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=\frac{a}{c}\) (c) If \(\operatorname{deg} f(x)>\operatorname{deg} g(x),\) then \(\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}\) does not exist.

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 \(f(c) ?\) $$\lim _{x \rightarrow 0} \tan \left(\frac{\pi}{2}-x\right)$$

Find the rule of the derivative of the function \(f .\) [See Example 7 and the remarks following it.] $$f(x)=3 x-5$$

Formal definitions of limits at infinity and negative infinity are given. Adapt the discussion in Special Topics \(13.2 .\) A to explain how these definitions are derived from the informal definitions given in this section. Let \(f\) be a function, and let \(L\) be a real number. Then the statement \(\lim f(x)=L\) means that for each positive number \(\epsilon\), there is a positive real number \(k\) (depending on \(\epsilon\) ) with this property: $$ \text { If } x>k . \text { then }|f(x)-L|<\epsilon $$ [Hint: Concentrate on the second part of the informal definition. The number \(k\) measures "large enough," that is, how large the values of \(x\) must be to guarantee that \(f(x)\) is as close as you want to \(L .]\)

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 \(f(c) ?\) $$\lim _{x \rightarrow 1} e^{x} \sin \frac{\pi x}{2}$$

Find the rule of the derivative of the function \(f .\) [See Example 7 and the remarks following it.] $$f(x)=x^{2}+x$$

Formal definitions of limits at infinity and negative infinity are given. Adapt the discussion in Special Topics \(13.2 .\) A to explain how these definitions are derived from the informal definitions given in this section. Let \(f\) be a function, and let \(L\) be a real number. Then the statement \(\lim _{x \rightarrow-\infty} f(x)=L\) means that for each positive number \(\epsilon,\) there is a negative real number \(n\) (depending on \(\epsilon\) ) with this property: $$ \text { If } x

(a) Approximate \(\lim _{x \rightarrow 0}(1+x)^{1 / x}\) to seven decimal places. (Evaluate the function at numbers closer and closer to 0 until successive approximations agree in the first seven decimal places.) (b) Find the decimal expansion of \(e\) to at least nine decimal places. (c) On the basis of the results of parts (a) and (b), what do you think is the exact value of \(\lim _{x \rightarrow 0}(1+x)^{1 / x} ?\)

Find the rule of the derivative of the function \(f .\) [See Example 7 and the remarks following it.] $$f(x)=x^{2}-x+1$$

(a) Graph the function \(f\) whose rule is $$f(x)=\left\\{\begin{array}{ll}3-x & \text { if } x<-2 \\\x+2 & \text { if }-2 \leq x<2 \\\1 & \text { if } x=2 \\\4-x & \text { if } x>2\end{array}\right.$$ Use the graph in part (a) to evaluate these limits: (b) \(\lim _{x \rightarrow-2} f(x)\) (c) \(\lim _{x \rightarrow 1} f(x)\) (d) \(\lim _{x \rightarrow 2} f(x)\)

Find the rule of the derivative of the function \(f .\) [The difference quotients of these functions were found and simplified in Exercises \(57-60 \text { of Section } 5.1 .]\) $$f(x)=\sqrt{x+1}$$

Find the rule of the derivative of the function \(f .\) [The difference quotients of these functions were found and simplified in Exercises \(57-60 \text { of Section } 5.1 .]\) $$f(x)=2 \sqrt{x+3}$$

(a) Graph the function \(g\) whose rule is $$g(x)=\left\\{\begin{array}{ll}x^{2} & \text { if } x<-1 \\ x+2 & \text { if }-1 \leq x<1 \\ 3-x & \text { if } x \geq 1\end{array}\right.$$ Use the graph in part (a) to evaluate these limits: (b) \(\lim _{x \rightarrow-1} g(x)\) (c) \(\lim _{x \rightarrow 0} g(x)\) (d) \(\lim _{x \rightarrow 1} g(x)\)

Find the rule of the derivative of the function \(f .\) [The difference quotients of these functions were found and simplified in Exercises \(57-60 \text { of Section } 5.1 .]\) $$f(x)=\sqrt{x^{2}+1}$$