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2.2E: Limits of Functions Exercises

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2.2: The Limit of a Function

Estimating limits from TABLES

For the following exercises, consider the function f(x)=x21x1 .

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 limx1f(x)? Explain your response.

Answer:

limx1f(x) does not exist because

limx1f(x)=2limx1+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: limx0(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] limx0sin(2x)x;±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
It appears that: limx0sin2xx=2

36) [T] limx0sin(3x)x±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.

37) Use the preceding two exercises to conjecture (guess) the value of the following limit: limx0sin(ax)x for a, a positive real value.

Answer:

It appears that: limx0sinaxx=a

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

38) limx2x24x2+x6

x x24x2+x6 x x24x2+x6
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) limx1(12x)

x 12x x 12x
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: limx1(12x)=1

40) limx051e1/x

x 51e1/x x 51e1/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) limz0z1z2(z+3)

z z1z2(z+3) z z1z2(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: limx0z1z2(z+3)=

42) limt0+cos(t)t

\(t) costt
0.1 a.
0.01 b.
0.001 c.
0.0001 d.

43) limx212xx24

x 12xx24 x 12xx24
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: limx212xx24=0.1250=18

[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

CNX_Calc_Figure_02_02_214.jpeg

 

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.

CNX_Calc_Figure_02_02_202.jpeg

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.

CNX_Calc_Figure_02_02_203.jpeg

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.

CNX_Calc_Figure_02_02_204.jpeg

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.

CNX_Calc_Figure_02_02_205.jpeg

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.

CNX_Calc_Figure_02_02_206.jpeg

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.

CNX_Calc_Figure_02_02_207.jpeg

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

CNX_Calc_Figure_02_02_209.jpeg

 

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

CNX_Calc_Figure_02_02_211.jpeg

 

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.

CNX_Calc_Figure_02_02_215.jpeg

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

This page titled 2.2E: Limits of Functions Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax.

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