Chapter Summary

In this chapter we:

•

defined the limit,
•

found accessible ways to approximate their values numerically and graphically,
•

developed a method of proving the value of a limit ( $\epsilon$ - $\delta$ proofs),
•

explored when limits do not exist,
•

considered limits that involved infinity, and
•

defined continuity and explored properties of continuous functions.

Why? Mathematics is famous for building on itself and calculus proves to be no exception. In the next chapter we will be interested in “dividing by 0.” That is, we will want to divide a quantity by a smaller and smaller number and see what value the quotient approaches. In other words, we will want to find a limit. These limits will enable us to, among other things, determine exactly how fast something is moving when we are only given position information.

Later, we will want to add up an infinite list of numbers. We will do so by first adding up a finite list of numbers, then take a limit as the number of things we are adding approaches infinity. Surprisingly, this sum often is finite; that is, we can add up an infinite list of numbers and get, for instance, 42.

These are just two quick examples of why we are interested in limits. Many students dislike this topic when they are first introduced to it, but over time an appreciation is often formed based on the scope of its applicability.

Exercises 1.6

Terms and Concepts

1.

In your own words, describe what it means for a function to be continuous.
2.

In your own words, describe what the Intermediate Value Theorem states.
3.

What is a “root” of a function?
4.

Given functions $f$ and $g$ on an interval $I$ , how can the Bisection Method be used to find a value $c$ where $f(c)=g(c)$ ?
5.

T/F: If $f$ is defined on an open interval containing $c$ , and $\displaystyle\lim_{x\to c}f(x)$ exists, then $f$ is continuous at $c$ .
6.

T/F: If $f$ is continuous at $c$ , then $\displaystyle\lim_{x\to c}f(x)$ exists.
7.

T/F: If $f$ is continuous at $c$ , then $\displaystyle\lim_{x\to c^{+}}f(x)=f(c)$ .
8.

T/F: If $f$ is continuous on $[a,b]$ , then $\displaystyle\lim_{x\to a^{-}}f(x)=f(a)$ .
9.

T/F: If $f$ is continuous on $[0,1)$ and $[1,2)$ , then $f$ is continuous on $[0,2)$ .
10.

T/F: The sum of continuous functions is also continuous.

Problems

In Exercises 11–18, a graph of a function $f$ is given along with a value $a$ . Determine if $f$ is continuous at $a$ ; if it is not, state why it is not.

11.
$a=1$
12.
$a=1$
13.
$a=1$
14.
$a=0$
15.
$a=1$
16.
$a=4$
17.
(a) $a=-2$ (b) $a=0$ (c) $a=2$
18.
$a=3\pi/2$

In Exercises 19–22, determine if $f$ is continuous at the indicated values. If not, explain why.

19.
$\displaystyle f(x)=\begin{cases}1&x=0\\ \frac{\sin x}{x}&x\neq 0\end{cases}$ (a) $x=0$ (b) $x=\pi$
20.
$\displaystyle f(x)=\begin{cases}x^{3}-x&x<1\\ x-2&x\geq 1\end{cases}$ (a) $x=0$ (b) $x=1$
21.
$\displaystyle f(x)=\begin{cases}\frac{x^{2}+5x+4}{x^{2}+3x+2}&x\neq-1\\ 3&x=-1\end{cases}$ (a) $x=-1$ (b) $x=10$
22.
$\displaystyle f(x)=\begin{cases}\frac{x^{2}-64}{x^{2}-11x+24}&x\neq 8\\ 5&x=8\end{cases}$ (a) $x=0$ (b) $x=8$

In Exercises 23–36, give the intervals on which the given function is continuous.

23.

$f(x)=x^{2}-3x+9$
24.

$\displaystyle g(x)=\sqrt{x^{2}-4}$
25.

$\displaystyle g(x)=\sqrt{4-x^{2}}$
26.

$\displaystyle h(k)=\sqrt{1-k}+\sqrt{k+1}$
27.

$\displaystyle f(t)=\sqrt{5t^{2}-30}$
28.

$\displaystyle g(t)=\frac{1}{\sqrt{1-t^{2}}}$
29.

$\displaystyle g(x)=\frac{1}{1+x^{2}}$
30.

$\displaystyle f(x)=e^{x}$
31.

$\displaystyle g(s)=\ln s$
32.

$\displaystyle h(t)=\cos t$
33.

$\displaystyle f(k)=\sqrt{1-e^{k}}$
34.

$\displaystyle f(x)=\sin(e^{x}+x^{2})$
35.

$\displaystyle f(x)=\begin{cases}\frac{x+1}{x+4}&x<2\\ x^{2}-3&2\leq x\leq 5\\ 6-2x&x>5\end{cases}$
36.

$\displaystyle f(x)=\begin{cases}\frac{1}{x-1}&x<0\\ 2x^{2}-3x-1&0\leq x\leq 2\\ 5x^{2}-4x&x>2\end{cases}$
37.
Let $\displaystyle f(x)=\begin{cases}x^{2}-1&x<3\\ x+5&x\geq 3\end{cases}$ . Is $f$ continuous everywhere?
38.
Let $\displaystyle f(x)=\begin{cases}x\sin(\frac{1}{x})&x\neq 0\\ 0&x=0\end{cases}$ . Is $f$ continuous everywhere?

Exercises 39–42 test your understanding of the Intermediate Value Theorem.

39.

Let $f$ be continuous on $[1,5]$ where $f(1)=-2$ and $f(5)=-10$ . Does a value $1<c<5$ exist such that $f(c)=-9$ ? Why/why not?
40.

Let $g$ be continuous on $[-3,7]$ where $g(0)=0$ and $g(2)=25$ . Does a value $-3<c<7$ exist such that $g(c)=15$ ? Why/why not?
41.

Let $f$ be continuous on $[-1,1]$ where $f(-1)=-10$ and $f(1)=10$ . Does a value $-1<c<1$ exist such that $f(c)=11$ ? Why/why not?
42.

Let $h$ be a function on $[-1,1]$ where $h(-1)=-10$ and $h(1)=10$ . Does a value $-1<c<1$ exist such that $h(c)=0$ ? Why/why not?

In Exercises 43–46, find the value(s) of $a$ and $b$ so that the function is continuous on $\mathbb{R}$ .

43.

$\displaystyle g(x)=\begin{cases}ax^{2}+3x&x<2\\ x^{3}-ax&x\geq 2\end{cases}$
44.

$\displaystyle f(x)=\begin{cases}a^{2}x-a&x>3\\ 4&x\leq 3\end{cases}$
45.

$\displaystyle f(x)=\begin{cases}ax-b&x<-1\\ 2x^{2}+3ax+b&-1\leq x<1\\ 4&x\geq 1\end{cases}$
46.

$\displaystyle f(x)=\begin{cases}x^{2}+2x&x\leq a\\ -1&x>a\end{cases}$

In Exercises 47–50, sketch the graph of a function that has the following properties.

47.

$f$ is discontinuous at 3, but continuous from the left at 3, and continuous elsewhere.
48.

$f$ is discontinuous at -1 and 2, but continuous from the right at -1 and continuous from the left at 2, and continuous elsewhere.
49.

$f$ has a jump discontinuity at -2 and an infinite discontinuity at 4 and is continuous elsewhere.
50.

$f$ has a removable discontinuity at 2, is continuous only from the left at 5, and is continuous elsewhere.

In Exercises 51–54, show that the functions have at least one real root.

51.

$f(x)=x^{2}+2x-4$
52.

$f(x)=\sin x-1/2$
53.

$f(x)=e^{x}-2$
54.

$f(x)=\cos x-\sin x$

Review

55.

Let $\displaystyle f(x)=\begin{cases}x^{2}-5&x<5\\ 5x&x\geq 5\end{cases}$ .

(a) $\displaystyle\lim_{x\to 5^{-}}f(x)$ (b) $\displaystyle\lim_{x\to 5^{+}}f(x)$ (c) $\displaystyle\lim_{x\to 5}f(x)$ (d) $f(5)$
56.
Numerically approximate the following limits: (a) $\displaystyle\lim_{x\to-4/5^{+}}\frac{x^{2}-8.2x-7.2}{x^{2}+5.8x+4}$ (b) $\displaystyle\lim_{x\to-4/5^{-}}\frac{x^{2}-8.2x-7.2}{x^{2}+5.8x+4}$
57.

Give an example of function $f(x)$ for which $\displaystyle\lim_{x\to 0}f(x)$ does not exist.