Chapter 7

Is \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}\) nite? If so, nd two consecutive integers, one smaller than this limit and the other larger. We will return to this limit later in the course.

Evaluate the following limits; then discuss \(\lim _{x \rightarrow \infty} b^{x}\) for \(b>0\). (a) \(\lim _{x \rightarrow \infty}(1.1)^{x}\) (b) \(\lim _{x \rightarrow \infty}(0.9)^{x}\) (c) \(\lim _{x \rightarrow 0}(1.1)^{x}\) (d) \(\lim _{x \rightarrow-\infty}(1.1)^{x}\) (e) \(\lim _{x \rightarrow-\infty}(0.9)^{x}\)

In Problems 2 and 3 , find the limit. $$ \lim _{x \rightarrow 3} \frac{2 x^{3}-8 x^{2}+5 x+3}{x-3} $$

Evaluate the limits. A graph may be useful. (a) \(\lim _{x \rightarrow 2^{-}} \frac{1}{x-2}\) (b) \(\lim _{x \rightarrow 2^{+}} \frac{1}{x-2}\) (c) \(\lim _{x \rightarrow 2} \frac{1}{x-2}\)

Find the following. (a) \(\lim _{x \rightarrow 4} \frac{1}{(x-4)^{2}}\) (b) \(\lim _{x \rightarrow 4} \frac{x+3}{(x-4)^{2}}\) (c) \(\lim _{x \rightarrow 4} \frac{x^{2}+16}{(x-4)^{2}}\) (d) \(\lim _{x \rightarrow 4^{+}} \frac{1}{(x-4)}\) (e) \(\lim _{x \rightarrow 4^{-}} \frac{1}{(x-4)}\) (f) \(\lim _{x \rightarrow 4} \frac{x^{2}-16}{(x-4)}\)

We can de ne \(\left(-\frac{1}{2}\right)^{n}\) for any positive integer \(n\), but not for every real number. For instance, \(\left(-\frac{1}{2}\right)^{1 / 2}=\sqrt{-\frac{1}{2}}\), which is not de ned in real numbers. We ll write \(\lim _{n \rightarrow \infty}\left(-\frac{1}{2}\right)^{n}=L\) if \(\left(-\frac{1}{2}\right)^{n}\) can be made arbitrarily close to \(L\) for all positive integers 4 suf ciently large. (a) Find \(\lim _{n \rightarrow \infty}\left(-\frac{1}{2}\right)^{n}\), where \(n\) takes on only positive integer values. (b) Which two of the following statements are true? Explain. i. \(\lim _{n \rightarrow \infty}(-2)^{n}=\infty\) ii. \(\lim _{n \rightarrow \infty}(-2)^{n}=-\infty\) iii. \(\lim _{n \rightarrow \infty}(-2)^{n}\) does not exist iv. \(\lim _{n \rightarrow \infty}-2^{n}=\infty\) v. \(\lim _{n \rightarrow \infty}-2^{n}=-\infty\)

n Problems 2 and 3 , find the limit. $$ \lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x} $$

Evaluate the limits. A graph may be useful. (a) \(\lim _{x \rightarrow 1^{-}} \frac{1}{(x-1)^{2}}\) (b) \(\lim _{x \rightarrow 1^{+}} \frac{1}{(x-1)^{2}}\) (c) \(\lim _{x \rightarrow 1} \frac{1}{(x-1)^{2}}\) (d) \(\lim _{x \rightarrow-1} \frac{1}{(x-1)^{2}}\)

Find \(\lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4}\).

Each of the following limits are of the form \(\lim _{x \rightarrow a} f(x)\). Evaluate the limit and sketch a graph of \(f\) on some interval including \(a\). Make it clear from your sketch whether or not \(f\) is de ned at \(a\). (a) \(\lim _{h \rightarrow-2} \frac{(h-3)(h+2)}{h+2}\) (b) \(\lim _{x \rightarrow 5} \frac{x^{2}-25}{x+5}\) (c) \(\lim _{x \rightarrow-5} \frac{x^{2}-25}{x+5}\) (d) \(\lim _{t \rightarrow 0} \frac{t^{2}+\pi t}{t}\) (e) \(\lim _{h \rightarrow 0} \frac{h k+h^{2}}{h}\), where \(k\) is a constant (f) \(\lim _{w \rightarrow 2} \frac{(w-3)(w+1)(w-2)}{3 w-6}\)

(a) Suppose you are interested in nding \(\lim _{x \rightarrow 2} f(x)\), where the function \(f(x)\) is explicitly given by a formula. What approaches might you take to investigate this limit? (b) Suppose you re now interested in nding \(\lim _{x \rightarrow \infty} f(x)\), where \(f(x)\) again is explicitly given by a formula. What approaches might you take to investigate this limit?

Evaluate the limits. A graph may be useful. \(f(x)=\frac{|x-3|}{x-3}\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 4} f(x)\) (c) \(\lim _{x \rightarrow 3^{+}} f(x)\) (d) \(\lim _{x \rightarrow 3^{-}} f(x)\) (c) \(\lim _{x \rightarrow 3} f(x)\)

Discussion Question. Consider \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}\). This is a very important limit. (a) What is your guess for this limit? You are not expected to guess the right answer. Once you complete part \((b)\) you will see that the problem is subtle. (b) Use a calculator or computer to investigate the limit, graphically and numerically. Give a revised estimate of this limit. We will return to this limit in Chapter 15 .

Evaluate the limits. A graph may be useful. \(f(x)=\left\\{\begin{array}{ll}3 x+4, & x<0 \\ 2 x+4, & x \geq 0\end{array}\right.\) (a) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow-2} f(x)\) (c) \(\lim _{x \rightarrow 0^{+}} f(x)\) (d) \(\lim _{x \rightarrow 0^{-}} f(x)\) (e) \(\lim _{x \rightarrow 0} f(x)\)

Each of the functions in Problems 5 through 10 is either continuous on \((-\infty, \infty)\) or has a point of discontinuity at some point \((s) x=a .\) Determine any point \((s)\) of discontinuity. Is the point of discontinuity removable? In other words, can the function be made continuous by defining or redefining the function at the point of discontinuity? $$ f(x)=\frac{x^{2}-4}{x+2} $$

\(f(x)=\frac{x}{2}+3 ; \lim _{x \rightarrow 2} f(x)\)

Evaluate the limits. A graph may be useful. \(f(x)=\left\\{\begin{array}{ll}\pi x+1, & x>0 \\ \pi x-1, & x \leq 0\end{array}\right.\) (a) \(\lim _{x \rightarrow \frac{1}{\pi}} f(x)\) (b) \(\lim _{x \rightarrow-\frac{1}{\pi}} f(x)\) (c) \(\lim _{x \rightarrow 0} f(x)\)

Each of the functions in Problems 5 through 10 is either continuous on \((-\infty, \infty)\) or has a point of discontinuity at some point \((s) x=a .\) Determine any point \((s)\) of discontinuity. Is the point of discontinuity removable? In other words, can the function be made continuous by defining or redefining the function at the point of discontinuity? $$ f(x)=\left\\{\begin{array}{ll} x^{3}, & x \neq 0 \\ 3, & x=0 \end{array}\right. $$

\(f(x)=\pi x-4\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 1} f(x)\) (c) \(\lim _{x \rightarrow \infty} f(x)\)

Evaluate the limits. A graph may be useful. \(f(x)=\left\\{\begin{array}{ll}x^{2}+3, & x \geq 1 \\ 2 x+2, & x<1\end{array}\right.\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 1} f(x)\) (c) \(\lim _{x \rightarrow 2} f(x)\)

Each of the functions in Problems 5 through 10 is either continuous on \((-\infty, \infty)\) or has a point of discontinuity at some point \((s) x=a .\) Determine any point \((s)\) of discontinuity. Is the point of discontinuity removable? In other words, can the function be made continuous by defining or redefining the function at the point of discontinuity? $$ f(x)=\frac{1}{x+2} $$

\(f(x)=|x-2| ;\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 2} f(x)\)

Let \(f\) be the function de ned by \(f(x)=\left\\{\begin{array}{ll}x+1, & \text { for } x \text { not an integer, } \\ 0, & \text { for } x \text { an integer. }\end{array}\right.\) (a) Sketch \(f\). (b) Find the following limits. i. \(\lim _{x \rightarrow 1.5} f(x)\) ii. \(\lim _{x \rightarrow 2} f(x)\) iii. \(\lim _{x \rightarrow 0} f(x)\) (c) For what values of \(c\) is \(\lim _{x \rightarrow c} f(x)=c+1 ?\) Have you excluded any values of \(c\) ? If so, which ones and why? Explain.

Each of the functions in Problems 5 through 10 is either continuous on \((-\infty, \infty)\) or has a point of discontinuity at some point \((s) x=a .\) Determine any point \((s)\) of discontinuity. Is the point of discontinuity removable? In other words, can the function be made continuous by defining or redefining the function at the point of discontinuity? $$ f(x)=\frac{1}{x^{2}+2} $$

\(f(x)=\frac{x^{2}-2 x}{x-2}\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 2} f(x)\)

Each of the functions in Problems 5 through 10 is either continuous on \((-\infty, \infty)\) or has a point of discontinuity at some point \((s) x=a .\) Determine any point \((s)\) of discontinuity. Is the point of discontinuity removable? In other words, can the function be made continuous by defining or redefining the function at the point of discontinuity? $$ f(x)=\left\\{\begin{array}{ll} -x^{2}-1, & x>0 \\ 5 x-1, & x<0 \end{array}\right. $$

\(f(x)=\left\\{\begin{array}{ll}5 x+1, & x \neq 2 \\ 7, & x=2\end{array}\right.\) (a) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 2} f(x)\)

Let \(f\) be the function given by $$ f(x)=\left\\{\begin{array}{ll} \frac{1}{x+7}, & \text { for } x \leq-5 \\ x+4, & \text { for }-52 \end{array}\right. $$ Evaluate the limits below. (a) \(\lim _{x \rightarrow-\infty} f(x)\) (b) \(\lim _{x \rightarrow-7} f(x)\) (c) \(\lim _{x \rightarrow-5^{+}} f(x)\) (d) \(\lim _{x \rightarrow-5} f(x)\) (e) \(\lim _{x \rightarrow-2} f(x)\) (f) \(\lim _{x \rightarrow 2^{+}} f(x)\) (g) \(\lim _{x \rightarrow \infty} f(x)\) (h) \(\lim _{x \rightarrow 0} f(x)\)

Each of the functions in Problems 5 through 10 is either continuous on \((-\infty, \infty)\) or has a point of discontinuity at some point \((s) x=a .\) Determine any point \((s)\) of discontinuity. Is the point of discontinuity removable? In other words, can the function be made continuous by defining or redefining the function at the point of discontinuity? $$ f(x)=\left\\{\begin{array}{ll} -x^{2}-x, & x>0 \\ 5 x-1, & x<0 \end{array}\right. $$

\(f(x)=\frac{(x+2)\left(x^{2}-x\right)}{x(x-1)}\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 1} f(x)\)

Let \(f(x)=\left\\{\begin{array}{ll}-x^{2}+1, & x>0, \\ a x+b, & x \leq 0 .\end{array}\right.\) What are the constraints on \(a\) and \(b\) in order for \(f\) to be continuous at \(x=0\) ?

\(f(x)=\frac{x^{2}-3 x-4}{x+1}\) (a) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow-1} f(x)\)

(a) Sketch the graph of \(f(x)=2|x-2|\). (b) What is \(f^{\prime}(0)\) ? What is \(f^{\prime}(4) ?\) What is \(f^{\prime}(2) ?\) (c) Find \(\lim _{h \rightarrow 0^{+}} \frac{f(2+h)-f(2)}{h}\) and \(\lim _{h \rightarrow 0^{-}} \frac{f(2+h)-f(2)}{h}\). Do your answers make sense to you? (d) On a separate set of axes graph \(f^{\prime}(x)\).

Let \(f(x)=\left\\{\begin{array}{l}g(x), x \geq a, \\ h(x), x

\(f(x)=\left\\{\begin{array}{ll}|x|, & x \neq 3 \\ 0, & x=3\end{array}\right.\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 3} f(x)\) (c) \(\lim _{x \rightarrow-\infty} f(x)\)

Let \(f(x)=\left\\{\begin{array}{ll}-1, & \text { if } x \text { is a rational number, } \\ 1, & \text { if } x \text { is an irrational number. }\end{array}\right.\) (a) Is \(f(x)\) continuous at \(x=1 ?\) (b) Is \(f(x)\) continuous at \(x=\pi\) ?

\(f(x)=\frac{x^{2}-9}{x+3}\) (a) \(\lim _{x \rightarrow 3} f(x)\) (b) \(\lim _{x \rightarrow-3} f(x)\)

\lim _{x \rightarrow 2} \frac{e^{x}-e^{2}}{x-2}

Let \(f(x)=\left\\{\begin{array}{ll}x^{2}, & \text { for } x \geq 0 \\ -x^{2}, & \text { for } x<0\end{array}\right.\) (a) Is \(f\) continuous at \(x=0 ?\) (b) Is \(f\) differentiable at \(x=0 ?\) If so, what is \(f^{\prime}(0)\) ?

$$ \lim _{h \rightarrow 0} \frac{\sqrt{9+h}-3}{h} $$

Find the derivative of \(f(x)=k x^{4}\), where \(k\) is a constant.

$$ \lim _{h \rightarrow 0} \frac{e^{1+h}-e}{h} $$

Let \(g(x)=\left\\{\begin{array}{ll}x^{2}, & \text { for } x \geq 0, \\ x, & \text { for } x<0\end{array}\right.\) (a) Is \(g\) continuous at \(x=0\) ? (b) Is \(g\) differentiable at \(x=0\) ? If so, what is \(g^{\prime}(0)\) ?

$$ \lim _{h \rightarrow 0} \frac{\sqrt{7+h}-\sqrt{7}}{h} $$

Let \(f(x)=\frac{1}{x}\). (a) Draw the graph of \(f(x)\) and \(f^{\prime}(x)\). (b) Use the graphs you ve drawn in part (a) to do the following. i. Find \(\lim _{x \rightarrow \infty} f^{\prime}(x)\). ii. Find \(\lim _{x \rightarrow 0^{+}} f^{\prime}(x)\). iii. Find \(\lim _{x \rightarrow 0^{-}} f^{\prime}(x)\). iv. Find \(\lim _{x \rightarrow-\infty} f^{\prime}(x)\). (c) Use the limit de nition of derivative to nd \(f^{\prime}(x) .\) Use your work to check your answers to parts (a) and (b).

$$ \lim _{h \rightarrow 0} \frac{e^{h}-1}{h} $$

The domain of a function \(f\) is all real numbers. The zeros of \(f(x)\) are \(x=-1, x=2\), and \(x=6\). There are no other \(x\) -values such that \(f(x)=0 .\) Is it possible that \(f(3)>0\) and \(f(4)<0 ?\) Explain.

In Problems 19 and 20, give an example of a function having the set of characteristics specified. (a) \(\lim _{x \rightarrow 5} f(x)=7 ; f(5)=7\) (b) \(\lim _{x \rightarrow 5} g(x)=7 ; g(5)=8\)

The domain of a continuous function \(f\) is all real numbers. The zeros of \(f\) are \(x=-1, x=2\), and \(x=6 .\) There are no other \(x\) -values such that \(f(x)=0 .\) Is it possible that \(f(3)>0\) and \(f(4)<0 ?\) Explain.

Give an example of a function having the set of characteristics specified. (a) \(\lim _{x \rightarrow \infty} f(x)=\infty ; \lim _{x \rightarrow-\infty} f(x)=-\infty ; \lim _{x \rightarrow 0} f(x)=1\) (b) \(\lim _{x \rightarrow \infty} g(x)=\infty ; \lim _{x \rightarrow-\infty} g(x)=\infty ; \lim _{x \rightarrow 0} g(x)=-1\)