12.E: Introduction to Calculus (Exercises)
 Page ID
 7390
12.1: Finding Limits  Numerical and Graphical Approaches
In this section, we will examine numerical and graphical approaches to identifying limits.
Verbal
1) Explain the difference between a value at \(x=a\) and the limit as \(x\) approaches \(a\).
 Answer

The value of the function, the output, at \(x=a\) is \(f(a)\). When the \(\lim \limits_{x \to a}f(x)\) is taken, the values of \(x\) get infinitely close to \(a\) but never equal \(a\). As the values of \(x\) approach \(a\) from the left and right, the limit is the value that the function is approaching.
2) Explain why we say a function does not have a limit as \(x\) approaches \(a\) if, as \(x\) approaches \(a\), the lefthand limit is not equal to the righthand limit.
Graphical
For the exercises 314, estimate the functional values and the limits from the graph of the function \(f\) provided in the Figure below.
3) \(\lim \limits_{x \to −2^−} f(x)\)
 Answer

\(4\)
4) \(\lim \limits_{x \to −2^+ }f(x)\)
5) \(\lim \limits_{x \to −2 f(x)}\)
 Answer

\(4\)
6) \(f(−2)\)
7) \(\lim \limits_{x \to −1^− f(x)}\)
 Answer

\(2\)
8) \(\lim \limits_{x \to 1^+} f(x)\)
9) \(\lim \limits_{x \to 1} f(x)\)
 Answer

does not exist
10) \(f(1)\)
11) \(\lim \limits_{x \to 4^−} f(x)\)
 Answer

\(4\)
12) \(\lim \limits_{x \to 4^+} f(x)\)
13) \(\lim \limits_{x \to 4} f(x)\)
 Answer

does not exist
14) \(f(4)\)
For the exercises 1521, draw the graph of a function from the functional values and limits provided.
15) \(\lim \limits_{x \to 0^−} f(x)=2, \lim \limits_{x \to 0^+} f(x)=–3, \lim \limits_{x \to 2} f(x)=2, f(0)=4, f(2)=–1, f(–3) \text{ does not exist.}\)
 Answer

Answers will vary.
16) \(\lim \limits_{x \to 2^−} f(x)=0,\lim \limits_{x \to 2^+} =–2,\lim \limits_{x \to 0} f(x)=3, f(2)=5, f(0)\)
 Answer

Answers will vary.
17) \(\lim \limits_{ x \to 2^−} f(x)=2, \lim \limits_{ x \to 2^+} f(x)=−3, \lim \limits_{x \to 0} f(x)=5, f(0)=1, f(1)=0\)
 Answer

Answers will vary.
18) \(\lim \limits_{x \to 3^−} f(x)=0, \lim \limits_{x \to 3^+} f(x)=5, \lim \limits_{x \to 5} f(x)=0, f(5)=4, f(3) \text{ does not exist.}\)
 Answer

Answers will vary.
19) \( \lim \limits_{ x \to 4} f(x)=6, \lim \limits_{ x \to 6^+} f(x)=−1, \lim \limits_{ x \to 0} f(x)=5, f(4)=6, f(2)=6\)
 Answer

Answers will vary.
20) \( \lim \limits_{ x \to −3} f(x)=2, \lim \limits_{ x \to 1^+} f(x)=−2, \lim \limits_{ x \to 3} f(x)=–4, f(–3)=0, f(0)=0\)
 Answer

Answers will vary.
21) \( \lim \limits_{ x \to π} f(x)=π^2, \lim \limits_{ x \to –π} f(x)=\dfrac{π}{2}, \lim \limits_{ x \to 1^} f(x)=0, f(π)=\sqrt{2}, f(0) \text{ does not exist}.\)
 Answer

Answers will vary.
For the exercises 2226, use a graphing calculator to determine the limit to \(5\) decimal places as \(x\) approaches \(0\).
22) \(f(x)=(1+x)^{\frac{1}{x}}\)
23) \(g(x)=(1+x)^{\frac{2}{x}}\)
 Answer

\(7.38906\)
24) \(h(x)=(1+x)^{\frac{3}{x}}\)
25) \(i(x)=(1+x)^{\frac{4}{x}}\)
 Answer

\(54.59815\)
26) \(j(x)=(1+x)^{\frac{5}{x}}\)
27) Based on the pattern you observed in the exercises above, make a conjecture as to the limit of \(f(x)=(1+x)^{\frac{6}{x}}, g(x)=(1+x)^{\frac{7}{x}},\) and \(h(x)=(1+x)^{\frac{n}{x}}.\)
 Answer

\(e^6≈403.428794,e^7≈1096.633158, e^n\)
For the exercises 2829, use a graphing utility to find graphical evidence to determine the left and righthand limits of the function given as \(x\) approaches \(a\). If the function has a limit as \(x\) approaches \(a\),state it. If not, discuss why there is no limit.
28) \((x)= \begin{cases} x−1, && \text{if }x≠1 \\ x^3, && \text{if }x=1 \end{cases} a=1 \)
29) \((x)= \begin{cases} \frac{1}{x+1}, && \text{if } x=−2 \\ (x+1)^2, && \text{if } x≠−2 \end{cases} a=−2 \)
 Answer

\(\lim \limits_{x \to −2} f(x)=1\)
Numeric
For the exercises 3038, use numerical evidence to determine whether the limit exists at \(x=a\). If not, describe the behavior of the graph of the function near \(x=a\). Round answers to two decimal places.
30) \(f(x)=\dfrac{x^2−4x}{16−x^2};a=4\)
31) \(f(x)=\dfrac{x^2−x−6}{x^2−9};a=3\)
 Answer

\(\lim \limits_{x \to 3} \left (\dfrac{x^2−x−6}{x^2−9} \right )=\dfrac{5}{6}≈0.83\)
32) \(f(x)=\dfrac{x^2−6x−7}{x^2– 7x};a=7\)
33) \(f(x)=\dfrac{x^2–1}{x^2–3x+2};a=1\)
 Answer

\(\lim \limits_{x \to 1} \left (\dfrac{x^2−1}{x^2−3x+2} \right )=−2.00\)
34) \(f(x)=\dfrac{1−x^2}{x^2−3x+2};a=1\)
35) \(f(x)=\dfrac{10−10x^2}{x^2−3x+2};a=1\)
 Answer

\(\lim \limits_{x \to 1} \left (\dfrac{10−10x^2}{x^2−3x+2} \right )=20.00\)
36) \(f(x)=\dfrac{x}{6x^2−5x−6};a=\dfrac{3}{2}\)
37) \(f(x)=\dfrac{x}{4x^2+4x+1};a=−\dfrac{1}{2}\)
 Answer

\(\lim \limits_{x \to \frac{−1}{2}} \left (\dfrac{x}{4x^2+4x+1} \right )\) does not exist. Function values decrease without bound as \(x\) approaches \(0.5\) from either left or right.
38) \(f(x)=\frac{2}{x−4}; a=4\)
For the exercises 3941, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as \(x\) approaches the given value.
39) \(\lim \limits_{x \to 0} \dfrac{7 \tan x}{3x}\)
 Answer

\(\lim \limits_{x \to 0} \dfrac{7 \tan x}{3x}=\dfrac{7}{3}\)
40) \(\lim \limits_{x \to 4} \dfrac{x^2}{x−4}\)
 Answer
41) \(\lim \limits_{x \to 0}\dfrac{2 \sin x}{4 \tan x}\)
 Answer

\(\lim \limits_{x \to 0} \dfrac{2 \sin x}{4 \tan x}=\dfrac{1}{2}\)
For the exercises 4249, use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as \(x\) approaches \(a\). If the function has a limit as \(x\) approaches \(a\), state it. If not, discuss why there is no limit.
42) \(\lim \limits_{x \to 0}e^{e^{\frac{1}{x}}}\)
43) \(\lim \limits_{x \to 0}e^{e^{− \frac{1}{x^2}}}\)
 Answer

\(\lim \limits_{x \to 0}e^{e^{− \frac{1}{x^2}}}=1.0\)
44) \(\lim \limits_{x \to 0} \dfrac{x}{x}\)
45) \(\lim \limits_{x \to −1} \dfrac{x+1}{x+1}\)
 Answer

\(\lim \limits_{ x→−1^−}\dfrac{ x+1 }{x+1}=\dfrac{−(x+1)}{(x+1)}=−1\) and \(\lim \limits_{ x \to −1^+}\dfrac{ x+1 }{x+1}=\dfrac{(x+1)}{(x+1)}=1\); since the righthand limit does not equal the lefthand limit, \(\lim \limits_{ x \to −1}\dfrac{x+1}{x+1}\) does not exist.
46) \(\lim \limits_{ x \to 5} \dfrac{ x−5 }{5−x}\)
47) \(\lim \limits_{ x \to −1}\dfrac{1}{(x+1)^2}\)
 Answer

\(\lim \limits_{ x \to −1} \dfrac{1}{(x+1)^2}\) does not exist. The function increases without bound as \(x\) approaches \(−1\) from either side.
48) \(\lim \limits_{ x \to 1} \dfrac{1}{(x−1)^3}\)
49) \(\lim \limits_{ x \to 0} \dfrac{5}{1−e^{\frac{2}{x}}}\)
 Answer

\(\lim \limits_{ x \to 0} \dfrac{5}{1−e^{\frac{2}{x}}}\) does not exist. Function values approach \(5\) from the left and approach \(0\) from the right.
50) Use numerical and graphical evidence to compare and contrast the limits of two functions whose formulas appear similar: \(f(x)=\left  \dfrac{1−x}{x} \right \) and \(g(x)=\left  \dfrac{1+x}{x} \right \) as \(x\) approaches \(0\). Use a graphing utility, if possible, to determine the left and righthand limits of the functions \(f(x)\) and \(g(x)\) as \(x\) approaches \(0\). If the functions have a limit as \(x\) approaches \(0\), state it. If not, discuss why there is no limit.
Extensions
51) According to the Theory of Relativity, the mass m m of a particle depends on its velocity \(v\). That is
\[m=\dfrac{m_o}{\sqrt{1−(v^2/c^2)}} \nonumber \]
where \(m_o\) is the mass when the particle is at rest and \(c\) is the speed of light. Find the limit of the mass, \(m\), as \(v\) approaches \(c^−.\)
 Answer

Through examination of the postulates and an understanding of relativistic physics, as \(v→c, m→∞. \)Take this one step further to the solution, \[\lim \limits_{v \to c^−}m=\lim \limits_{v \to c^−} \dfrac{m_o}{\sqrt{1−(v^2/c^2)}}=∞ \nonumber \]
52) Allow the speed of light, \(c\), to be equal to \(1.0\). If the mass, \(m\), is \(1\), what occurs to \(m\) as \(v \to c\)? Using the values listed in the Table below, make a conjecture as to what the mass is as \(v\) approaches \(1.00\).
\(v\)  \(m\) 

0.5  1.15 
0.9  2.29 
0.95  3.20 
0.99  7.09 
0.999  22.36 
0.99999  223.61 
12.2: Finding Limits  Properties of Limits
Graphing a function or exploring a table of values to determine a limit can be cumbersome and timeconsuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits. Knowing the properties of limits allows us to compute limits directly.
Verbal
1) Give an example of a type of function \(f\) whose limit, as \(x\) approaches \(a,\) is \(f(a)\).
 Answer

If \(f\) is a polynomial function, the limit of a polynomial function as \(x\) approaches \(a\) will always be \(f(a)\).
2) When direct substitution is used to evaluate the limit of a rational function as \(x\) approaches \(a\) and the result is \(f(a)=\dfrac{0}{0}\),does this mean that the limit of \(f\) does not exist?
3) What does it mean to say the limit of \(f(x)\), as \(x\) approaches \(c\), is undefined?
 Answer

It could mean either (1) the values of the function increase or decrease without bound as \(x\) approaches \(c,\) or (2) the left and righthand limits are not equal.
Algebraic
For the exercises 430, evaluate the limits algebraically.
4) \(\lim \limits_{x \to 0} (3)\)
5) \(\lim \limits_{x \to 2} \left (\dfrac{−5x}{x^2−1} \right )\)
 Answer

\(\dfrac{−10}{3}\)
6) \(\lim \limits_{x \to 2} \left (\dfrac{x^2−5x+6}{x+2} \right )\)
7) \(\lim \limits_{x \to 3} \left (\dfrac{x^2−9}{x−3} \right )\)
 Answer

\(6\)
8) \(\lim \limits_{x \to −1} \left (\dfrac{x^2−2x−3}{x+1} \right )\)
9) \(\lim \limits_{x \to \frac{3}{2}} \left (\dfrac{6x^2−17x+12}{2x−3} \right )\)
 Answer

\(\dfrac{1}{2}\)
10) \(\lim \limits_{ x \to −\frac{7}{2}} \left (\dfrac{8x^2+18x−35}{2x+7} \right )\)
11) \(\lim \limits_{ x \to 3} \left (\dfrac{x^2−9}{x−5x+6} \right )\)
 Answer

\(6\)
12) \(\lim \limits_{ x \to −3} \left (\dfrac{−7x^4−21x^3}{−12x^4+108x^2} \right )\)
13) \(\lim \limits_{ x \to 3} \left (\dfrac{x^2+2x−3}{x−3} \right )\)
 Answer

does not exist
14) \(\lim \limits_{ h \to 0} \left (\dfrac{(3+h)^3−27}{h} \right )\)
15) \(\lim \limits_{ h \to 0} \left (\dfrac{(2−h)^3−8}{h} \right )\)
 Answer

\(−12\)
16) \(\lim \limits_{ h \to 0} \left (\dfrac{(h+3)^2−9}{h} \right )\)
17) \(\lim \limits_{ h \to 0} \left (\dfrac{\sqrt{5−h}−\sqrt{5}}{h} \right )\)
 Answer

\(−\dfrac{\sqrt{5}}{10}\)
18) \(\lim \limits_{ x \to 0} \left (\dfrac{\sqrt{3−x}−\sqrt{3}}{x} \right )\)
19) \(\lim \limits_{ x \to 9} \left (\dfrac{x^2−81}{3−x} \right )\)
 Answer

\(−108\)
20) \(\lim \limits_{ x \to 1} \left (\dfrac{\sqrt{x}−x^2}{1−\sqrt{x}} \right )\)
21) \(\lim \limits_{ x \to 0}\left ( \dfrac{x}{\sqrt{1+2x}1} \right )\)
 Answer

\(1\)
22) \(\lim \limits_{ x \to \frac{1}{2}} \left (\dfrac{x^2−\tfrac{1}{4}}{2x−1} \right )\)
23) \(\lim \limits_{ x \to 4} \left (\dfrac{x^3−64}{x^2−16} \right )\)
 Answer

\(6\)
24) \(\lim \limits_{ x \to 2^−} \left (\dfrac{x−2}{x−2} \right )\)
25) \(\lim \limits_{ x \to 2^+} \left (\dfrac{ x−2 }{x−2} \right )\)
 Answer

\(1\)
26) \(\lim \limits_{ x \to 2} \left (\dfrac{ x−2 }{x−2} \right )\)
27) \(\lim \limits_{ x \to 4^−} \left (\dfrac{ x−4 }{4−x} \right )\)
 Answer

\(1\)
28) \(\lim \limits_{ x \to 4^+} \left (\dfrac{ x−4 }{4−x} \right )\)
29) \(\lim \limits_{ x \to 4} \left (\dfrac{ x−4 }{4−x} \right )\)
 Answer

does not exist
30) \(\lim \limits_{ x \to 2} \left (\dfrac{−8+6x−x^2}{x−2} \right )\)
For the exercises 3133, use the given information to evaluate the limits: \(\lim \limits_{x \to c}f(x)=3, \lim \limits_{x \to c} g(x)=5\)
31) \(\lim \limits_{x \to c} [ 2f(x)+\sqrt{g(x)} ]\)
 Answer

\(6+\sqrt{5}\)
32) \(\lim \limits_{x \to c} [ 3f(x)+\sqrt{g(x)} ]\)
33) \(\lim \limits_{x \to c}\dfrac{f(x)}{g(x)}\)
 Answer

\(\dfrac{3}{5}\)
For the exercises 3443, evaluate the following limits.
34) \(\lim \limits_{x \to 2} \cos (πx)\)
35) \(\lim \limits_{x \to 2} \sin (πx)\)
 Answer

\(0\)
36) \(\lim \limits_{x \to 2} \sin \left (\dfrac{π}{x} \right )\)
37) \(f(x)= \begin{cases} 2x^2+2x+1, && x≤0 \\ x−3, && x>0 ; \end{cases} \lim \limits_{x \to 0^+}f(x)\)
 Answer

\(−3\)
38) \(f(x)= \begin{cases} 2x^2+2x+1, && x≤0 \\ x−3, && x>0 ; \end{cases} \lim \limits_{x \to 0^−} f(x)\)
39) \(f(x)= \begin{cases} 2x^2+2x+1, && x≤0 \\ x−3, && x>0 ; \end{cases} \lim \limits_{x \to 0}f(x)\)
 Answer

does not exist; righthand limit is not the same as the lefthand limit.
40) \(\lim \limits_{x \to 4} \dfrac{\sqrt{x+5}−3}{x−4}\)
41) \(\lim \limits_{x \to 2^+} (2x−〚x〛)\)
 Answer

\(2\)
42) \(\lim \limits_{x \to 2} \dfrac{\sqrt{x+7}−3}{x^2−x−2}\)
43) \(\lim \limits_{x \to 3^+}\dfrac{x^2}{x^2−9}\)
 Answer

Limit does not exist; limit approaches infinity.
For the exercises 4453, find the average rate of change\(\dfrac{f(x+h)−f(x)}{h}\).
44) \(f(x)=x+1\)
45) \(f(x)=2x^2−1\)
 Answer

\(4x+2h\)
46) \(f(x)=x^2+3x+4\)
47) \(f(x)=x^2+4x−100\)
 Answer

\(2x+h+4\)
48) \(f(x)=3x^2+1\)
49) \(f(x)= \cos (x)\)
 Answer

\(\dfrac{\cos (x+h)− \cos (x)}{h}\)
50) \(f(x)=2x^3−4x\)
51) \(f(x)=\dfrac{1}{x}\)
 Answer

\(\dfrac{−1}{x(x+h)}\)
52) \(f(x)=\dfrac{1}{x^2}\)
53) \(f(x)=\sqrt{x}\)
 Answer

\(\dfrac{−1}{\sqrt{x+h}+\sqrt{x}}\)
Graphical
54) Find an equation that could be represented by the Figure below.
 Answer

\(f(x)=\dfrac{x^2+5x+6}{x+3}\)
For the exercises 5657, refer to the Figure below.
56) What is the righthand limit of the function as \(x\) approaches \(0\)?
57) What is the lefthand limit of the function as \(x\) approaches \(0\)?
 Answer

does not exist
RealWorld Applications
58) The position function \(s(t)=−16t^2+144t\) gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval \([ 1,2 ]\).
59) The height of a projectile is given by \(s(t)=−64t^2+192t\) Find the average rate of change of the height from \(t=1\) second to \(t=1.5\) seconds.
 Answer

\(52\)
60) The amount of money in an account after \(t\) years compounded continuously at \(4.25\%\) interest is given by the formula \(A=A_0e^{0.0425t}\),where \(A_0\) is the initial amount invested. Find the average rate of change of the balance of the account from \(t=1\) year to \(t=2\) years if the initial amount invested is \(\$1,000.00.\)
12.3: Continuity
A function that remains level for an interval and then jumps instantaneously to a higher value is called a stepwise function. This function is an example. A function that has any hole or break in its graph is known as a discontinuous function. A stepwise function, such as parkinggarage charges as a function of hours parked, is an example of a discontinuous function. We can check three different conditions to decide if a function is continuous at a particular number.
Verbal
1) State in your own words what it means for a function \(f\) to be continuous at \(x=c\).
 Answer

Informally, if a function is continuous at \(x=c\), then there is no break in the graph of the function at \(f(c)\), and \(f(c)\) is defined.
2) State in your own words what it means for a function to be continuous on the interval \((a,b)\).
Algebraic
For the exercises 322, determine why the function \(f\) is discontinuous at a given point \(a\) on the graph. State which condition fails.
3) \(f(x)=\ln  x+3 ,a=−3\)
 Answer

discontinuous at \(a=−3\); \(f(−3)\) does not exist
4) \(f(x)= \ln  5x−2 ,a=\dfrac{2}{5}\)
5) \(f(x)=\dfrac{x^2−16}{x+4},a=−4\)
 Answer

removable discontinuity at \(a=−4; f(−4)\) is not defined
6) \(f(x)=\dfrac{x^2−16x}{x},a=0\)
7) \(f(x)= \begin{cases} x, && x≠3 \\ 2x, && x=3 \end{cases} a=3\)
 Answer

Discontinuous at \(a=3; \lim \limits_{x \to 3} f(x)=3,\) but \(f(3)=6,\) which is not equal to the limit.
8) \(f(x) = \begin{cases} 5, &&x≠0 \\ 3, && x=0 \end{cases} a=0\)
9) \(f(x)= \begin{cases} \dfrac{1}{2−x}, && x≠2 \\ 3, &&x=2 \end{cases} a=2\)
 Answer

\(\lim \limits_{x \to 2}f(x)\) does not exist.
10) \(f(x)= \begin{cases} \dfrac{1}{x+6}, && x=−6 \\ x^2, && x≠−6 \end{cases} a=−6\)
11) \(f(x)=\begin{cases} 3+x, &&x<1 \\ x, &&x=1 \\ x^2, && x>1 \end{cases} a=1\)
 Answer

\(\lim \limits_{x \to 1^−}f(x)=4;\lim \limits_{x \to 1^+}f(x)=1.\) Therefore, \(\lim \limits_{x \to 1}f(x)\) does not exist.
12) \(f(x)= \begin{cases} 3−x, && x<1 \\ x, && x=1 \\ 2x^2, && x>1 \end{cases} a=1\)
13) \(f(x)= \begin{cases} 3+2x, && x<1 \\ x, && x=1 \\ −x^2, && x>1 \end{cases} a=1\)
 Answer

\(\lim \limits_{x \to 1^−} f(x)=5≠ \lim \limits_{x \to 1^+}f(x)=−1\). Thus \(\lim \limits_{x \to 1}f(x)\) does not exist.
14) \(f(x)= \begin{cases} x^2, &&x<−2 \\ 2x+1, && x=−2 \\ x^3, && x>−2 \end{cases} a=−2\)
15) \(f(x)= \begin{cases} \dfrac{x^2−9}{x+3}, && x<−3 \\ x−9, && x=−3 \\ \dfrac{1}{x}, && x>−3 \end{cases} a=−3\)
 Answer

\(\lim \limits_{x to −3^+}f(x)=−\dfrac{1}{3}\)
Therefore, \(\lim \limits_{x \to −3} f(x)\) does not exist.
16) \(f(x)= \begin{cases} \dfrac{x^2−9}{x+3}, && x<−3 \\ x−9, && x=−3\\ −6, && x>−3 \end{cases} a=3\)
17) \(f(x)=\dfrac{x^2−4}{x−2}, a=2\)
 Answer

\(f(2)\) is not defined.
18) \(f(x)=\dfrac{25−x^2}{x^2−10x+25}, a=5\)
19) \(f(x)=\dfrac{x^3−9x}{x^2+11x+24}, a=−3\)
 Answer

\(f(−3)\) is not defined.
20) \(f(x)=\dfrac{x^3−27}{x^2−3x}, a=3\)
21) \(f(x)=\dfrac{x}{x}, a=0\)
 Answer

\(f(0)\) is not defined.
22) \(f(x)=\dfrac{2x+2}{x+2}, a=−2\)
For the exercises 2335, determine whether or not the given function \(f\) is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.
23) \(f(x)=x^3−2x−15\)
 Answer

Continuous on \((−∞,∞)\)
24) \(f(x)=\dfrac{x^2−2x−15}{x−5}\)
25) \(f(x)=2⋅3^{x+4}\)
 Answer

Continuous on \((−∞,∞)\)
26) \(f(x)=− \sin (3x)\)
27) \(f(x)=\dfrac{x−2}{x^2−2x}\)
 Answer

Discontinuous at \(x=0\) and\(x=2\)
28) \(f(x)= \tan (x)+2\)
29) \(f(x)=2x+\dfrac{5}{x}\)
 Answer

Discontinuous at \(x=0\)
30) \(f(x)=\log _2 (x)\)
31) \(f(x)= \ln x^2 \)
 Answer

Continuous on \((0,∞)\)
32) \(f(x)=e^{2x}\)
33) \(f(x)=\sqrt{x−4}\)
 Answer

Continuous on \([4,∞)\)
34) \(f(x)= \sec (x)−3\)
35) \(f(x)=x^2+ \sin (x)\)
 Answer

Continuous on \((−∞,∞)\).
36) Determine the values of \(b\) and \(c\) such that the following function is continuous on the entire real number line.
\[f(x)= \begin{cases}x+1, && 1<x<3 \\ x^2+bx+c, &&x−2≥1 \end{cases} \nonumber \]
Graphical
For the exercises 3739, refer to the Figure below. Each square represents one square unit. For each value of \(a\), determine which of the three conditions of continuity are satisfied at \(x=a\) and which are not.
37) \(x=−3\)
 Answer

\(1\), but not \(2\) or \(3\)
38) \(x=2\)
39) \(x=4\)
 Answer

\(1\) and \(2\), but not \(3\)
For the exercises 4043, use a graphing utility to graph the function \(f(x)= \sin \left (\dfrac{12π}{x} \right )\) as in Figure. Set the \(x\)axis a short distance before and after \(0\) to illustrate the point of discontinuity.
40) Which conditions for continuity fail at the point of discontinuity?
41) Evaluate \(f(0)\).
 Answer

\(f(0)\) is undefined.
42) Solve for \(x\) if \(f(x)=0\).
43) What is the domain of \(f(x)\)?
 Answer

\((−∞,0)∪(0,∞)\)
For the exercises 4445, consider the function shown in the Figure below.
44) At what \(x\)coordinates is the function discontinuous?
45) What condition of continuity is violated at these points?
 Answer

At \(x=−1\), the limit does not exist. At \(x=1, f(1)\) does not exist.
At \(x=2\), there appears to be a vertical asymptote, and the limit does not exist.
46) Consider the function shown in the Figure below. At what \(x\)coordinates is the function discontinuous? What condition(s) of continuity were violated?
47) Construct a function that passes through the origin with a constant slope of \(1\), with removable discontinuities at \(x=−7\) and \(x=1\).
 Answer

\(\dfrac{x^3+6x^2−7x}{(x+7)(x−1)}\)
48) The function \(f(x)=\dfrac{x^3−1}{x−1}\) is graphed in the Figure below. It appears to be continuous on the interval \([−3,3]\), but there is an \(x\)value on that interval at which the function is discontinuous. Determine the value of \(x\) at which the function is discontinuous, and explain the pitfall of utilizing technology when considering continuity of a function by examining its graph.
49) Find the limit \(\lim \limits_{ x \to 1}f(x)\) and determine if the following function is continuous at \(x=1\):
\[fx= \begin{cases} x^2+4 && x≠1 \\ 2 && x=1\end{cases} \nonumber \]
 Answer

The function is discontinuous at \(x=1\) because the limit as \(x\) approaches \(1\) is \(5\) and \(f(1)=2\).
50) The graph of \(f(x)= \dfrac{\sin (2x)}{x}\) is shown in the Figure below. Is the function \(f(x)\) continuous at \(x=0?\) Why or why not?
12.4: Derivatives
Change divided by time is one example of a rate. The rates of change in the previous examples are each different. In other words, some changed faster than others. If we were to graph the functions, we could compare the rates by determining the slopes of the graphs.
Verbal
1) How is the slope of a linear function similar to the derivative?
 Answer

The slope of a linear function stays the same. The derivative of a general function varies according to \(x\). Both the slope of a line and the derivative at a point measure the rate of change of the function.
2) What is the difference between the average rate of change of a function on the interval \([x,x+h]\) and the derivative of the function at