2.2E: Exercises for Section 2.2
( \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 lim? Explain your response.
- Answer:
-
\displaystyle \lim_{x \to 1}f(x) does not exist because \displaystyle \lim_{x \to 1^−}f(x)=−2≠\lim_{x \to 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:
- \displaystyle \lim_{x \to 0}(1+x)^{1/x}\approx 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] \displaystyle \lim_{x \to 0}\frac{\sin 2x}{x};\quad ±0.1,\; ±0.01, \; ±0.001, \;±.0001
x | \frac{\sin 2x}{x} | x | \frac{\sin 2x}{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. |
- 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;
\displaystyle \lim_{x \to 0}\frac{\sin 2x}{x}=2
7) [T] \displaystyle \lim_{x \to 0}\frac{\sin 3x}{x};\quad ±0.1, \; ±0.01, \; ±0.001, \; ±0.0001
x | \frac{\sin 3x}{x} | x | \frac{\sin 3x}{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. |
8) Use the preceding two exercises to conjecture (guess) the value of the following limit: \displaystyle \lim_{x \to 0}\frac{\sin ax}{x} for a, a positive real value.
- Answer:
- \displaystyle \lim_{x \to 0}\frac{\sin ax}{x}=a
[T] In exercises 9 - 12, set up a table of values to find the indicated limit. Round to eight digits.
9) \displaystyle \lim_{x \to 2}\frac{x^2−4}{x^2+x−6}
x | \frac{x^2−4}{x^2+x−6} | x | \frac{x^2−4}{x^2+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) \displaystyle \lim_{x \to 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;
\displaystyle \lim_{x \to 1}(1−2x)=−1
11) \displaystyle \lim_{x \to 0}\frac{5}{1−e^{1/x}}
x | \frac{5}{1−e^{1/x}} | x | \frac{5}{1−e^{1/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) \displaystyle \lim_{x \to 2}\frac{1−\frac{2}{x}}{x^2−4}
x | \frac{1−\frac{2}{x}}{x^2−4} | x | \frac{1−\frac{2}{x}}{x^2−4} |
---|---|---|---|
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}
In exercises 13 - 17, use the following graph of the function y=f(x) to find the values, if possible. Estimate when necessary.
13) \displaystyle \lim_{x→1^−}f(x)
14) \displaystyle \lim_{x→1^+}f(x)
- Answer:
- 2
15) \displaystyle \lim_{x→1}f(x)
16) \displaystyle \lim_{x→2}f(x)
- Answer:
- 1
17) f(1)
In exercises 18 - 21, use the graph of the function y=f(x) shown here to find the values, if possible. Estimate when necessary.
18) \displaystyle \lim_{x→0^−}f(x)
- Answer:
- 1
19) \displaystyle \lim_{x→0^+}f(x)
20) \displaystyle \lim_{x→0}f(x)
- Answer:
- DNE
21) \displaystyle \lim_{x→2}f(x)
In exercises 22 - 27, use the graph of the function y=f(x) shown here to find the values, if possible. Estimate when necessary.
22) \displaystyle \lim_{x→−2^−}f(x)
- Answer:
- 0
23) \displaystyle \lim_{x→−2^+}f(x)
24) \displaystyle \lim_{x→−2}f(x)
- Answer:
- DNE
25) \displaystyle \lim_{x→2^−}f(x)
26) \displaystyle \lim_{x→2^+}f(x)
- Answer:
- 2
27) \displaystyle \lim_{x→2}f(x)
In exercises 28 - 30, use the graph of the function y=g(x) shown here to find the values, if possible. Estimate when necessary.
28) \displaystyle \lim_{x→0^−}g(x)
- Answer:
- 3
29) \displaystyle \lim_{x→0^+}g(x)
30) \displaystyle \lim_{x→0}g(x)
- Answer:
- DNE
In exercises 31 - 33, use the graph of the function y=h(x) shown here to find the values, if possible. Estimate when necessary.
31) \displaystyle \lim_{x→0^−}h(x)
32) \displaystyle \lim_{x→0^+}h(x)
- Answer:
- 0
33) \displaystyle \lim_{x→0}h(x)
In exercises 34 - 38, use the graph of the function y=f(x) shown here to find the values, if possible. Estimate when necessary.
34) \displaystyle \lim_{x→0^−}f(x)
- Answer:
- -2
35) \displaystyle \lim_{x→0^+}f(x)
36) \displaystyle \lim_{x→0}f(x)
- Answer:
- DNE
37) \displaystyle \lim_{x→1}f(x)
38) \displaystyle \lim_{x→2}f(x)
- Answer:
- 0
39) 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 |
40) 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.
Contributors and Attributions
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.