# 2.3E: Exercises for Section 2.2


## Intuitive Definition of Limits

For 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 $$\displaystyle \lim_{x→1}f(x)$$? Explain your response.

$$\displaystyle \lim_{x \to 1}f(x)$$ 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 $$f(x)=(1+x)^{1/x}$$?

$$\displaystyle \lim_{x \to 0}(1+x)^{1/x}\approx 2.7183$$.

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.
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] $$\displaystyle \lim_{x \to 0}\frac{\sin 3x}{x} ±0.1, \; ±0.01, \; ±0.001, \; ±0.0001$$

$$x$$ $$\frac{\sin 3x}{x}$$ $$x$$ $$\frac{\sin 3x}{x}$$
-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.

$$\displaystyle \lim_{x \to 0}\frac{\sin ax}{x}=a$$

[T] In exercises 9 - 14, set up a table of values to find the indicated limit. Round to eight significant 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.
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) $$\displaystyle \lim_{z \to 0}\frac{z−1}{z^2(z+3)}$$

$$z$$ $$\frac{z−1}{z^2(z+3)}$$ $$z$$ $$\frac{z−1}{z^2(z+3)}$$
-0.1 a. 0.1 e.
-0.01 b. 0.01 f.
-0.001 c. 0.001 g.
-0.0001 d. 0.0001 h.
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
$$\displaystyle \lim_{x \to 0}\frac{z−1}{z^2(z+3)}=−∞$$

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

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

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

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

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) $$\displaystyle \lim_{x→−2^+}f(x)=3$$

False; $$\displaystyle \lim_{x→−2^+}f(x)=+∞$$

19) $$\displaystyle \lim_{x→−8}f(x)=f(−8)$$

20) $$\displaystyle \lim_{x→6}f(x)=5$$

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) $$\displaystyle \lim_{x→1^+}f(x)$$

$$2$$

23) $$\displaystyle \lim_{x→1}f(x)$$

24) $$\displaystyle \lim_{x→2}f(x)$$

$$1$$

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) $$\displaystyle \lim_{x→0^−}f(x)$$

$$1$$

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

28) $$\displaystyle \lim_{x→0}f(x)$$

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) $$\displaystyle \lim_{x→−2^−}f(x)$$

$$0$$

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

32) $$\displaystyle \lim_{x→−2}f(x)$$

DNE

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

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

$$2$$

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) $$\displaystyle \lim_{x→0^−}g(x)$$

$$3$$

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

38) $$\displaystyle \lim_{x→0}g(x)$$

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) $$\displaystyle \lim_{x→0^+}h(x)$$

$$0$$

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) $$\displaystyle \lim_{x→0^−}f(x)$$

$$-2$$

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

44) $$\displaystyle \lim_{x→0}f(x)$$

DNE

45) $$\displaystyle \lim_{x→1}f(x)$$

46) $$\displaystyle \lim_{x→2}f(x)$$

$$0$$

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

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

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.

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

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