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# 2.2E: Limits of Functions Exercises


## 2.2: The Limit of a Function

### Estimating limits from TABLES

For the following exercises, consider the function $$f(x)=\frac{∣x^2−1∣}{x−1}$$ .

30) [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.

31) What do your results in the preceding exercise indicate about the two-sided limit $$\displaystyle \lim_{x→1}f(x)$$? Explain your response.

Answer:

$$\displaystyle \lim_{x \to 1}f(x)$$ does not exist because

$$\displaystyle \lim_{x \to 1^−}f(x)=−2$$ ≠ $$\displaystyle \lim_{x \to 1^+}f(x)=2$$.

For the following exercises, consider the function $$f(x)=(1+x)^{1/x}$$.

32) [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.

33) What does the table of values in the preceding exercise indicate about the function $$f(x)=(1+x)^{1/x}$$?

Answer:

It appears that: $$\displaystyle \lim_{x \to 0}(1+x)^{1/x}=2.7183$$

34) To which mathematical constant does the limit in the preceding exercise appear to be getting closer?

In the following exercises, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.

35) [T] $$\displaystyle \lim_{x \to 0}\frac{\sin(2x)}{x};±0.1,±0.01,±0.001,±.0001$$

$$x$$ $$\frac{sin2x}{x}$$ $$x$$ $$\frac{sin2x}{x}$$
-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
It appears that: $$\displaystyle \lim_{x \to 0}\frac{sin2x}{x}=2$$

36) [T] $$\displaystyle \lim_{x \to 0}\frac{\sin(3x)}{x} ±0.1, ±0.01, ±0.001, ±0.0001$$

$$x$$ $$\frac{sin3x}{x}$$ $$x$$ $$\frac{sin3x}{x}$$
-0.1 a. 0.1 e.
-0.01 b. 0.01 f.
-0.001 c. 0.001 g.
-0.0001 d. 0.0001 h.

37) 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:

It appears that: $$\displaystyle \lim_{x \to 0}\frac{sinax}{x}=a$$

[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.

38) $$\displaystyle \lim_{x \to 2}\frac{x^2−4}{x^2+x−6}$$

$$x$$ $$\frac{x^2−4}{x^2+x−6}$$ $$x$$ $$\frac{x^2−4}{x^2+x−6}$$
1.9 a. 2.1 e.
1.99 b. 2.01 f.
1.999 c. 2.001 g.
1.9999 d. 2.0001 h.

39) $$\displaystyle \lim_{x \to 1}(1−2x)$$

$$x$$ $$1−2x$$ $$x$$ $$1−2x$$
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;

It appears that: $$\displaystyle \lim_{x \to 1}(1−2x)=−1$$

40) $$\displaystyle \lim_{x \to 0}\frac{5}{1−e^{1/x}}$$

$$x$$ $$\frac{5}{1−e^{1/x}}$$ $$x$$ $$\frac{5}{1−e^{1/x}}$$
-0.1 a. 0.1 e.
-0.01 b. 0.01 f.
-0.001 c. 0.001 g.
-0.0001 d. 0.0001 h.

41) $$\displaystyle \lim_{z \to 0}\frac{z−1}{z^2(z+3)}$$

$$z$$ $$\frac{z−1}{z^2(z+3)}$$ $$z$$ $$\frac{z−1}{z^2(z+3)}$$
-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. −37.931934
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

It appears that: $$\lim_{x \to 0}\frac{z−1}{z^2(z+3)}=−∞$$

42) $$\displaystyle \lim_{t \to 0^+}\frac{\cos(t)}{t}$$

$$t) \(\frac{cost}{t}$$
0.1 a.
0.01 b.
0.001 c.
0.0001 d.

43) $$\displaystyle \lim_{x \to 2}\frac{1−\frac{2}{x}}{x^2−4}$$

$$x$$ $$\frac{1−\frac{2}{x}}{x^2−4}$$ $$x$$ $$\frac{1−\frac{2}{x}}{x^2−4}$$
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

It appears that: $$\displaystyle \lim_{x \to 2}\frac{1−\frac{2}{x}}{x^2−4}=0.1250=\frac{1}{8}$$

[T] In the following exercises, 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?

44) $$\displaystyle \lim_{θ \to 0}sin(\frac{π}{θ})$$

$$θ$$ $$sin(\frac{π}{θ})$$ $$θ$$ $$sin(\frac{π}{θ})$$
-0.1 a. 0.1 e.
-0.01 b. 0.01 f.
-0.001 c. 0.001 g.
-0.0001 d. 0.0001 h.

45) $$\displaystyle \lim_{α \to 0^+} \frac{1}{α}\cos(\frac{π}{α})$$

$$a$$ $$\frac{1}{α}cos(\frac{π}{α})$$
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

It appears from the table: $$\displaystyle \lim_{α→0^+}\frac{1}{α}\cos(\frac{π}{α})=-∞$$
actual answer (see graph): DNE

### Estimating limits from GRAPHS

Note: Exercises #46 - #49 moved to section 2.4 Exercises

In the following exercises, use the following graph of the function $$y=f(x)$$ to find the values, if possible. Estimate when necessary.

50) $$\displaystyle \lim_{x→1^−}f(x)$$

51) $$\displaystyle \lim_{x→1^+}f(x)$$

Answer:

2

52) $$\displaystyle \lim_{x→1}f(x)$$

53) $$\displaystyle \lim_{x→2}f(x)$$

Answer:

1

54) $$f(1)$$

In the following exercises, use the graph of the function $$y=f(x)$$ shown here to find the values, if possible. Estimate when necessary.

55) $$\displaystyle \lim_{x→0^−}f(x)$$

Answer:

1

56) $$\displaystyle \lim_{x→0^+}f(x)$$

57) $$\displaystyle \lim_{x→0}f(x)$$

Answer:

DNE

58) $$\displaystyle \lim_{x→2}f(x)$$

In the following exercises, use the graph of the function $$y=f(x)$$ shown here to find the values, if possible. Estimate when necessary.

59) $$\displaystyle \lim_{x→−2^−}f(x)$$

Answer:

0

60) $$\displaystyle \lim_{x→−2^+}f(x)$$

61) $$\displaystyle \lim_{x→−2}f(x)$$

Answer:

DNE

62) $$\displaystyle \lim_{x→2^−}f(x)$$

63) $$\displaystyle \lim_{x→2^+}f(x)$$

Answer:

2

64) $$\displaystyle \lim_{x→2}f(x)$$

In the following exercises, use the graph of the function $$y=g(x)$$ shown here to find the values, if possible. Estimate when necessary.

65) $$\displaystyle \lim_{x→0^−}g(x)$$

Answer:

3

66) $$\displaystyle \lim_{x→0^+}g(x)$$

67) $$\displaystyle \lim_{x→0}g(x)$$

Answer:

DNE

In the following exercises, use the graph of the function $$y=h(x)$$ shown here to find the values, if possible. Estimate when necessary.

68) $$\displaystyle \lim_{x→0^−}h(x)$$

69) $$\displaystyle \lim_{x→0^+}h(x)$$

Answer:

0

70) $$\displaystyle \lim_{x→0}h(x)$$

In the following exercises, use the graph of the function $$y=f(x)$$ shown here to find the values, if possible. Estimate when necessary.

71) $$\displaystyle \lim_{x→0^−}f(x)$$

Answer:

−2

72) $$\displaystyle \lim_{x→0^+}f(x)$$

73) $$\displaystyle \lim_{x→0}f(x)$$

Answer:

DNE

74) $$\displaystyle \lim_{x→1}f(x)$$

75) $$\displaystyle \lim_{x→2}f(x)$$

Answer:

0

In the following exercises, sketch the graph of a function with the given properties.

76) $$\displaystyle \lim_{x→2}f(x)=1, \displaystyle \lim_{x→4^−}f(x)=3, \displaystyle \lim_{x→4^+}f(x)=6,x=4$$ is not defined.

77) $$\displaystyle \lim_{x→−∞}f(x)=0, \displaystyle \lim_{x→−1^−}f(x)=−∞, \displaystyle \lim_{x→−1^+}f(x)=∞, \displaystyle \lim_{x→0}f(x)=f(0), f(0)=1, \displaystyle \lim_{x→∞}f(x)=−∞$$

Answer:

Answer may vary

78) $$\displaystyle \lim_{x→−∞}f(x)=2, \displaystyle \lim_{x→3^−}f(x)=−∞, \displaystyle \lim_{x→3^+}f(x)=∞, \displaystyle \lim_{x→∞}f(x)=2,f(0)=\frac{−1}{3}$$

79) $$\displaystyle \lim_{x→−∞}f(x)=2, \displaystyle \lim_{x→−2}f(x)=−∞, \displaystyle \lim_{x→∞} f(x)=2,f(0)=0$$

Answer:

Answer may vary

80)$$\displaystyle \lim_{x→−∞}f(x)=0, \displaystyle \lim_{x→−1^−}f(x)=∞, \displaystyle \lim_{x→−1^+}f(x)=−∞, f(0)=−1, \displaystyle \lim_{x→1^−}f(x)=−∞, \displaystyle \lim_{x→1^+}f(x)=∞, \displaystyle \lim_{x→∞}f(x)=0$$

81) 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 xSF 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 xSF 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.

82) 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