2.2E: Exercises for Section 2.2
- Last updated
- May 26, 2022
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- 102695
- Zoya Kravets
- Mission College
( \newcommand{\kernel}{\mathrm{null}\,}\)
Intuitive Definition of Limits
For exercises 1 - 2, consider the function f(x)=x2−1|x−1|.
1) [T] Complete the following table for the function. Round your solutions to four decimal places.
x | f(x) | x | f(x) |
---|---|---|---|
0.9 | a. | 1.1 | e. |
0.99 | b. | 1.01 | f. |
0.999 | c. | 1.001 | g. |
0.9999 | d. | 1.0001 | h. |
2) What do your results in the preceding exercise indicate about the two-sided limit limx→1f(x)? Explain your response.
- Answer
-
limx→1f(x) does not exist because limx→1−f(x)=−2≠limx→1+f(x)=2.
For exercises 3 - 5, consider the function f(x)=(1+x)1/x.
3) [T] Make a table showing the values of f for x=−0.01,−0.001,−0.0001,−0.00001 and for x=0.01,0.001,0.0001,0.00001. Round your solutions to five decimal places.
x | f(x) | x | f(x) |
---|---|---|---|
-0.01 | a. | 0.01 | e. |
-0.001 | b. | 0.001 | f. |
-0.0001 | c. | 0.0001 | g. |
-0.00001 | d. | 0.00001 | h. |
4) What does the table of values in the preceding exercise indicate about the function f(x)=(1+x)1/x?
- Answer
- limx→0(1+x)1/x≈2.7183.
5) To which mathematical constant do the values in the preceding exercise appear to be approaching? This is the actual limit here.
In exercises 6 - 8, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.
6) [T] limx→0sin2xx;±0.1,±0.01,±0.001,±.0001
x | sin2xx | x | sin2xx |
---|---|---|---|
-0.1 | a. | 0.1 | e. |
-0.01 | b. | 0.01 | f. |
-0.001 | c. | 0.001 | g. |
-0.0001 | d. | 0.0001 | h. |
- Answer
- a. 1.98669331; b. 1.99986667; c. 1.99999867; d. 1.99999999; e. 1.98669331; f. 1.99986667; g. 1.99999867; h. 1.99999999;
limx→0sin2xx=2
7) [T] limx→0sin3xx±0.1,±0.01,±0.001,±0.0001
x | sin3xx | x | sin3xx |
---|---|---|---|
-0.1 | a. | 0.1 | e. |
-0.01 | b. | 0.01 | f. |
-0.001 | c. | 0.001 | g. |
-0.0001 | d. | 0.0001 | h. |
8) Use the preceding two exercises to conjecture (guess) the value of the following limit: limx→0sinaxx for a, a positive real value.
- Answer
- limx→0sinaxx=a
[T] In exercises 9 - 14, set up a table of values to find the indicated limit. Round to eight significant digits.
9) limx→2x2−4x2+x−6
x | x2−4x2+x−6 | x | x2−4x2+x−6 |
---|---|---|---|
1.9 | a. | 2.1 | e. |
1.99 | b. | 2.01 | f. |
1.999 | c. | 2.001 | g. |
1.9999 | d. | 2.0001 | h. |
10) limx→1(1−2x)
x | 1−2x | x | 1−2x |
---|---|---|---|
0.9 | a. | 1.1 | e. |
0.99 | b. | 1.01 | f. |
0.999 | c. | 1.001 | g. |
0.9999 | d. | 1.0001 | h. |
- Answer
- a. −0.80000000; b. −0.98000000; c. −0.99800000; d. −0.99980000; e. −1.2000000; f. −1.0200000; g. −1.0020000; h. −1.0002000;
limx→1(1−2x)=−1
11) limx→051−e1/x
x | 51−e1/x | x | 51−e1/x |
---|---|---|---|
-0.1 | a. | 0.1 | e. |
-0.01 | b. | 0.01 | f. |
-0.001 | c. | 0.001 | g. |
-0.0001 | d. | 0.0001 | h. |
12) limz→0z−1z2(z+3)
z | z−1z2(z+3) | z | z−1z2(z+3) |
---|---|---|---|
-0.1 | a. | 0.1 | e. |
-0.01 | b. | 0.01 | f. |
-0.001 | c. | 0.001 | g. |
-0.0001 | d. | 0.0001 | h. |
- Answer
- a. −37.931034; b. −3377.9264; c. −333,777.93; d. −33,337,778; e. −29.032258; f. −3289.0365; g. −332,889.04; h. −33,328,889
limx→0z−1z2(z+3)=−∞
13) limt→0+costt
t | costt |
---|---|
0.1 | a. |
0.01 | b. |
0.001 | c. |
0.0001 | d. |
14) limx→21−2xx2−4
x | 1−2xx2−4 | x | 1−2xx2−4 |
---|---|---|---|
1.9 | a. | 2.1 | e. |
1.99 | b. | 2.01 | f. |
1.999 | c. | 2.001 | g. |
1.9999 | d. | 2.0001 | h. |
- Answer
- a. 0.13495277; b. 0.12594300; c. 0.12509381; d. 0.12500938; e. 0.11614402; f. 0.12406794; g. 0.12490631; h. 0.12499063;
\displaystyle ∴\lim_{x \to 2}\frac{1−\frac{2}{x}}{x^2−4}=0.1250=\frac{1}{8}
[T] In exercises 15 - 16, set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail?
15) \displaystyle \lim_{θ \to 0}\sin\left(\frac{π}{θ}\right)
θ | \sin\left(\frac{π}{θ}\right) | θ | \sin\left(\frac{π}{θ}\right) |
---|---|---|---|
-0.1 | a. | 0.1 | e. |
-0.01 | b. | 0.01 | f. |
-0.001 | c. | 0.001 | g. |
-0.0001 | d. | 0.0001 | h. |
16) \displaystyle \lim_{α \to 0^+} \frac{1}{α}\cos\left(\frac{π}{α}\right)
a | \frac{1}{α}\cos\left(\frac{π}{α}\right) |
---|---|
0.1 | a. |
0.01 | b. |
0.001 | c. |
0.0001 | d. |
- Answer
-
a. 10.00000; b. 100.00000; c. 1000.0000; d. 10,000.000;
Guess: \displaystyle \lim_{α→0^+}\frac{1}{α}\cos\left(\frac{π}{α}\right)=∞;
Actual: DNE , since the graph shows the function oscillates wildly between values approaching positive infinity and values approaching negative infinity, as the value of α approaches 0 from the positive side.
In exercises 17 - 20, consider the graph of the function y=f(x) shown here. Which of the statements about y=f(x) are true and which are false? Explain why a statement is false.
17) \displaystyle \lim_{x→10}f(x)=0
18) \displaystyle \lim_{x→−2^+}f(x)=3
- Answer
- False; \displaystyle \lim_{x→−2^+}f(x)=+∞
19) \displaystyle \lim_{x→−8}f(x)=f(−8)
20) \displaystyle \lim_{x→6}f(x)=5
- Answer
- False; \displaystyle \lim_{x→6}f(x) DNE since \displaystyle \lim_{x→6^−}f(x)=2 and \displaystyle \lim_{x→6^+}f(x)=5.
In exercises 21 - 25, use the following graph of the function y=f(x) to find the values, if possible. Estimate when necessary.
21) \displaystyle \lim_{x→1^−}f(x)
22) \displaystyle \lim_{x→1^+}f(x)
- Answer
- 2
23) \displaystyle \lim_{x→1}f(x)
24) \displaystyle \lim_{x→2}f(x)
- Answer
- 1
25) f(1)
In exercises 26 - 29, use the graph of the function y=f(x) shown here to find the values, if possible. Estimate when necessary.
26) \displaystyle \lim_{x→0^−}f(x)
- Answer
- 1
27) \displaystyle \lim_{x→0^+}f(x)
28) \displaystyle \lim_{x→0}f(x)
- Answer
- DNE
29) \displaystyle \lim_{x→2}f(x)
In exercises 30 - 35, use the graph of the function y=f(x) shown here to find the values, if possible. Estimate when necessary.
30) \displaystyle \lim_{x→−2^−}f(x)
- Answer
- 0
31) \displaystyle \lim_{x→−2^+}f(x)
32) \displaystyle \lim_{x→−2}f(x)
- Answer
- DNE
33) \displaystyle \lim_{x→2^−}f(x)
34) \displaystyle \lim_{x→2^+}f(x)
- Answer
- 2
35) \displaystyle \lim_{x→2}f(x)
In exercises 36 - 38, use the graph of the function y=g(x) shown here to find the values, if possible. Estimate when necessary.
36) \displaystyle \lim_{x→0^−}g(x)
- Answer
- 3
37) \displaystyle \lim_{x→0^+}g(x)
38) \displaystyle \lim_{x→0}g(x)
- Answer
- DNE
In exercises 39 - 41, use the graph of the function y=h(x) shown here to find the values, if possible. Estimate when necessary.
39) \displaystyle \lim_{x→0^−}h(x)
40) \displaystyle \lim_{x→0^+}h(x)
- Answer
- 0
41) \displaystyle \lim_{x→0}h(x)
In exercises 42 - 46, use the graph of the function y=f(x) shown here to find the values, if possible. Estimate when necessary.
42) \displaystyle \lim_{x→0^−}f(x)
- Answer
- -2
43) \displaystyle \lim_{x→0^+}f(x)
44) \displaystyle \lim_{x→0}f(x)
- Answer
- DNE
45) \displaystyle \lim_{x→1}f(x)
46) \displaystyle \lim_{x→2}f(x)
- Answer
- 0
Infinite Limits
In exercises 47 - 51, sketch the graph of a function with the given properties.
47) \displaystyle \lim_{x→2}f(x)=1, \quad \lim_{x→4^−}f(x)=3, \quad \lim_{x→4^+}f(x)=6, \quad x=4 is not defined.
48) \displaystyle \lim_{x→−∞}f(x)=0, \quad \lim_{x→−1^−}f(x)=−∞, \quad \lim_{x→−1^+}f(x)=∞,\quad \lim_{x→0}f(x)=f(0), \quad f(0)=1, \quad \lim_{x→∞}f(x)=−∞
- Answer
-
Answers may vary
49) \displaystyle \lim_{x→−∞}f(x)=2, \quad \lim_{x→3^−}f(x)=−∞, \quad \lim_{x→3^+}f(x)=∞, \quad \lim_{x→∞}f(x)=2, \quad f(0)=-\frac{1}{3}
50) \displaystyle \lim_{x→−∞}f(x)=2,\quad \lim_{x→−2}f(x)=−∞,\quad \lim_{x→∞}f(x)=2,\quad f(0)=0
- Answer
-
Answer may vary
51) \displaystyle \lim_{x→−∞}f(x)=0,\quad \lim_{x→−1^−}f(x)=∞,\quad \lim_{x→−1^+}f(x)=−∞, \quad f(0)=−1, \quad \lim_{x→1^−}f(x)=−∞, \quad \lim_{x→1^+}f(x)=∞, \quad \lim_{x→∞}f(x)=0
52) Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, x, is shown here. We are mainly interested in the location of the front of the shock, labeled X_{SF} in the diagram.
a. Evaluate \displaystyle \lim_{x→X_{SF}^+}ρ(x).
b. Evaluate \displaystyle \lim_{x→X_{SF}^−}ρ(x).
c. Evaluate \displaystyle \lim_{x→X_{SF}}ρ(x). Explain the physical meanings behind your answers.
- Answer
- a. ρ_2 b. ρ_1 c. DNE unless ρ_1=ρ_2. As you approach X_{SF} from the right, you are in the high-density area of the shock. When you approach from the left, you have not experienced the “shock” yet and are at a lower density.
53) A track coach uses a camera with a fast shutter to estimate the position of a runner with respect to time. A table of the values of position of the athlete versus time is given here, where x is the position in meters of the runner and t is time in seconds. What is \displaystyle \lim_{t→2}x(t)? What does it mean physically?
t(sec) | x(m) |
---|---|
1.75 | 4.5 |
1.95 | 6.1 |
1.99 | 6.42 |
2.01 | 6.58 |
2.05 | 6.9 |
2.25 | 8.5 |