Problem Set: The Limit of a Function

For the following exercises (1-2), consider the function $f(x)=\dfrac{x^2-1}{|x-1|}$ .

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 $\underset{x\to 1}{\lim}f(x)$ ? Explain your response.

Show Solution

$\underset{x\to 1}{\lim}f(x)$ does not exist because $\underset{x\to 1^-}{\lim}f(x)=-2 \ne \underset{x\to 1^+}{\lim}f(x)=2$ .

For the following 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 $f(x)=(1+x)^{1/x}$ ?

Show Solution

$\underset{x\to 0}{\lim}(1+x)^{1/x}=2.7183$

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

In the following exercises (6-8), use the given values of $x$ to set up a table to evaluate the limits. Round your solutions to eight decimal places.

6. [T] $\underset{x\to 0}{\lim}\dfrac{\sin 2x}{x}; \, x = \pm 0.1, \, \pm 0.01, \, \pm 0.001, \, \pm 0.0001$

$x$	$\frac{\sin 2x}{x}$	$x$	$\frac{\sin 2x}{x}$
−0.1	a.	0.1	e.
−0.01	b.	0.01	f.
−0.001	c.	0.001	g.
−0.0001	d.	0.0001	h.

Show Solution

a. 1.98669331; b. 1.99986667; c. 1.99999867; d. 1.99999999; e. 1.98669331; f. 1.99986667; g. 1.99999867; h. 1.99999999; $\underset{x\to 0}{\lim}\frac{\sin 2x}{x}=2$

7. [T] $\underset{x\to 0}{\lim}\dfrac{\sin 3x}{x}; \, x = \pm 0.1, \, \pm 0.01, \, \pm 0.001, \, \pm 0.0001$

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: $\underset{x\to 0}{\lim}\dfrac{\sin ax}{x}$ for $a$ , a positive real value.

Show Solution

$\underset{x\to 0}{\lim}\frac{\sin ax}{x}=a$

In the following exercises (9-14), set up a table of values to find the indicated limit. Round to eight digits.

9. [T] $\underset{x\to 2}{\lim}\dfrac{x^2-4}{x^2+x-6}$

$x$	$\frac{x^2-4}{x^2+x-6}$	$x$	$\frac{x^2-4}{x^2+x-6}$
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. [T] $\underset{x\to 1}{\lim}(1-2x)$

$x$	$1-2x$	$x$	$1-2x$
0.9	a.	1.1	e.
0.99	b.	1.01	f.
0.999	c.	1.001	g.
0.9999	d.	1.0001	h.

Show Solution

a. −0.80000000; b. −0.98000000; c. −0.99800000; d. −0.99980000; e. −1.2000000; f. −1.0200000; g. −1.0020000; h. −1.0002000;

$\underset{x\to 1}{\lim}(1-2x)=-1$

11. [T] $\underset{x\to 0}{\lim}\dfrac{5}{1-e^{1/x}}$

$x$	$\frac{5}{1-e^{1/x}}$	$x$	$\frac{5}{1-e^{1/x}}$
−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. [T] $\underset{z\to 0}{\lim}\dfrac{z-1}{z^2(z+3)}$

$z$	$\frac{z-1}{z^2(z+3)}$	$z$	$\frac{z-1}{z^2(z+3)}$
−0.1	a.	0.1	e.
−0.01	b.	0.01	f.
−0.001	c.	0.001	g.
−0.0001	d.	0.0001	h.

Show Solution

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

$\underset{x\to 0}{\lim}\frac{z-1}{z^2(z+3)}=−\infty$

13. [T] $\underset{t\to 0^+}{\lim}\dfrac{\cos t}{t}$

$t$	$\frac{\cos t}{t}$
0.1	a.
0.01	b.
0.001	c.
0.0001	d.

14. [T] $\underset{x\to 2^-}{\lim}\dfrac{1-\frac{2}{x}}{x^2-4}$

$x$	$\frac{1-\frac{2}{x}}{x^2-4}$	$x$	$\frac{1-\frac{2}{x}}{x^2-4}$
1.9	a.	2.1	e.
1.99	b.	2.01	f.
1.999	c.	2.001	g.
1.9999	d.	2.0001	h.

Show Solution

a. 0.13495277; b. 0.12594300; c. 0.12509381; d. 0.12500938; e. 0.11614402; f. 0.12406794; g. 0.12490631; h. 0.12499063;

$\underset{x\to 2^-}{\lim}\frac{1-\frac{2}{x}}{x^2-4}=0.1250=\frac{1}{8}$

In the following 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. [T] $\underset{\theta \to 0^-}{\lim}\sin \left(\frac{\pi }{\theta }\right)$

θ	$\sin (\frac{\pi }{\theta })$	θ	$\sin (\frac{\pi }{\theta })$
−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. [T] $\underset{\alpha \to 0^+}{\lim}\frac{1}{\alpha} \cos \left(\frac{\pi }{\alpha }\right)$

$a$	$\frac{1}{\alpha } \cos (\frac{\pi }{\alpha })$
0.1	a.
0.01	b.
0.001	c.
0.0001	d.

Show Solution

a. −10.00000; b. −100.00000; c. −1000.0000; d. −10,000.000; Guess: $\underset{\alpha \to 0^+}{\lim}\frac{1}{\alpha } \cos (\frac{\pi }{\alpha })=\infty$ , Actual: DNE

A graph of the function (1/alpha) * cos (pi / alpha), which oscillates gently until the interval [-.2, .2], where it oscillates rapidly, going to infinity and negative infinity as it approaches the y axis.

In the following 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. $\underset{x\to 10^-}{\lim}f(x)=0$

18. $\underset{x\to -2^+}{\lim}f(x)=3$

Show Solution

False; $\underset{x\to -2^+}{\lim}f(x)=+\infty$

19. $\underset{x\to -8^+}{\lim}f(x)=f(-8)$

20. $\underset{x\to 6}{\lim}f(x)=5$

Show Solution

False; $\underset{x\to 6}{\lim}f(x)$ DNE since $\underset{x\to 6^-}{\lim}f(x)=2$ and $\underset{x\to 6^+}{\lim}f(x)=5$ .

In the following exercises (21-24), use the following graph of the function $y=f(x)$ to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first segment exists for x <=1, and the second segment exists for x > 1. The first segment is linear with a slope of 1 and goes through the origin. Its endpoint is a closed circle at (1,1). The second segment is also linear with a slope of -1. It begins with the open circle at (1,2).

21. $\underset{x\to 1^-}{\lim}f(x)$

22. $\underset{x\to 1^+}{\lim}f(x)$

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23. $\underset{x\to 1}{\lim}f(x)$

24.

$\underset{x\to 2}{\lim}f(x)$

25.

$f(1)$

In the following exercises (26-29), use the graph of the function $y=f(x)$ shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first is a linear function for x < 0. There is an open circle at (0,1), and its slope is -1. The second segment is the right half of a parabola opening upward. Its vertex is a closed circle at (0, -4), and it goes through the point (2,0).

26. $\underset{x\to 0^-}{\lim}f(x)$

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27. $\underset{x\to 0^+}{\lim}f(x)$

28. $\underset{x\to 0}{\lim}f(x)$

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29. $\underset{x\to 2}{\lim}f(x)$

In the following exercises (30-35), use the graph of the function $y=f(x)$ shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with three segments, all linear. The first exists for x < -2, has a slope of 1, and ends at the open circle at (-2, 0). The second exists over the interval [-2, 2], has a slope of -1, goes through the origin, and has closed circles at its endpoints (-2, 2) and (2,-2). The third exists for x>2, has a slope of 1, and begins at the open circle (2,2).

30. $\underset{x\to -2^-}{\lim}f(x)$

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31. $\underset{x\to -2^+}{\lim}f(x)$

32. $\underset{x\to -2}{\lim}f(x)$

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33. $\underset{x\to 2^-}{\lim}f(x)$

34. $\underset{x\to 2^+}{\lim}f(x)$

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35. $\underset{x\to 2}{\lim}f(x)$

In the following exercises (36-38), use the graph of the function $y=g(x)$ shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first exists for x>=0 and is the left half of an upward opening parabola with vertex at the closed circle (0,3). The second exists for x>0 and is the right half of a downward opening parabola with vertex at the open circle (0,0).

36. $\underset{x\to 0^-}{\lim}g(x)$

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37. $\underset{x\to 0^+}{\lim}g(x)$

38. $\underset{x\to 0}{\lim}g(x)$

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In the following exercises (39-41), use the graph of the function $y=h(x)$ shown here to find the values, if possible. Estimate when necessary.

A graph of a function with two curves approaching 0 from quadrant 1 and quadrant 3. The curve in quadrant one appears to be the top half of a parabola opening to the right of the y axis along the x axis with vertex at the origin. The curve in quadrant three appears to be the left half of a parabola opening downward with vertex at the origin.

39. $\underset{x\to 0^-}{\lim}h(x)$

40. $\underset{x\to 0^+}{\lim}h(x)$

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41. $\underset{x\to 0}{\lim}h(x)$

In the following exercises (42-46), use the graph of the function $y=f(x)$ shown here to find the values, if possible. Estimate when necessary.

A graph with a curve and a point. The point is a closed circle at (0,-2). The curve is part of an upward opening parabola with vertex at (1,-1). It exists for x > 0, and there is a closed circle at the origin.

42. $\underset{x\to 0^-}{\lim}f(x)$

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43. $\underset{x\to 0^+}{\lim}f(x)$

44. $\underset{x\to 0}{\lim}f(x)$

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45. $\underset{x\to 1^-}{\lim}f(x)$

46. $\underset{x\to 2^+}{\lim}f(x)$

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In the following exercises (47-51), sketch the graph of a function with the given properties.

47. $\underset{x\to 2}{\lim}f(x)=1, \, \underset{x\to 4^-}{\lim}f(x)=3$ , $\underset{x\to 4^+}{\lim}f(x)=6, \, f(4)$ is not defined.

48. $\underset{x\to -\infty }{\lim}f(x)=0, \, \underset{x\to -1^-}{\lim}f(x)=−\infty$ , $\underset{x\to -1^+}{\lim}f(x)=\infty, \, \underset{x\to 0}{\lim}f(x)=f(0), \, f(0)=1, \, \underset{x\to \infty }{\lim}f(x)=−\infty$

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49. $\underset{x\to -\infty}{\lim}f(x)=2, \, \underset{x\to 3^-}{\lim}f(x)=−\infty$ , $\underset{x\to 3^+}{\lim}f(x)=\infty, \, \underset{x\to \infty }{\lim}f(x)=2, \, f(0)=-\frac{1}{3}$

50. $\underset{x\to -\infty }{\lim}f(x)=2, \, \underset{x\to -2}{\lim}f(x)=−\infty$ , $\underset{x\to \infty }{\lim}f(x)=2, \, f(0)=0$

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51. $\underset{x\to -\infty }{\lim}f(x)=0, \, \underset{x\to -1^-}{\lim}f(x)=\infty, \, \underset{x\to -1^+}{\lim}f(x)=−\infty, \, f(0)=-1, \, \underset{x\to 1^-}{\lim}f(x)=−\infty, \, \underset{x\to 1^+}{\lim}f(x)=\infty, \, \underset{x\to \infty }{\lim}f(x)=0$

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 graph in quadrant one of the density of a shockwave with three labeled points: p1 and p2 on the y axis, with p1 > p2, and xsf on the x axis. It consists of y= p1 from 0 to xsf, x = xsf from y= p1 to y=p2, and y=p2 for values greater than or equal to xsf.

a. Evaluate $\underset{x\to x_{SF}^+}{\lim}\rho(x)$

b. Evaluate $\underset{x\to x_{SF}^-}{\lim}\rho(x)$

c. Evaluate $\underset{x\to x_{SF}}{\lim}\rho(x)$ . Explain the physical meanings behind your answers.

Show Solution

a. $\rho_{2}$

b. $\rho_{1}$

c. DNE unless $\rho_{1}=\rho_{2}$ $ρ_{1} = ρ_{2} .$

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 $\underset{t\to 2}{\lim}x(t)$ ? 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

Limits: Supplemental Content

Problem Set: The Limit of a Function

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