2.2E: Exercises

Reading Questions

1. What is the intuitive meaning of the statement \( \displaystyle \lim_{x \to a} f(x) = L\)?
2. When evaluating \( \displaystyle \lim_{x \to a} f(x)\) numerically using a table, why is it important that \(x\) approaches \(a\) but \(x \neq a\)?
3. Describe the general process of using a table of values to estimate the limit of a function.
4. What are the three issues the text mentions regarding the use of numerical approximation (tables/technology) to state the value of a limit?
5. What does the notation \( \displaystyle \lim_{x \to a^-} f(x)\) represent?
6. What does the notation \( \displaystyle \lim_{x \to a^+} f(x)\) represent?
7. How is the existence and value of a two-sided limit, \( \displaystyle \lim_{x \to a} f(x)\), related to its corresponding one-sided limits?
8. What does it mean for a function to have an infinite limit, such as \( \displaystyle \lim_{x \to a} f(x) = \infty\)? According to the text, does this mean the limit "exists" in the same way a finite limit does?
9. What does "DNE" signify when discussing limits?
10. Why is it crucial to use radians instead of degrees when dealing with trigonometric functions in calculus limits?
11. State the "Limit of the Identity Function" and the "Limit of a Constant" as presented in the "Two Important Limits" theorem.
12. What is the primary warning given at the end of the section regarding the use of tables to estimate limits in future calculus work?

Homework

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

1) 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 \(\displaystyle \lim_{x \to 1}f(x)\)? Explain your response.

For exercises 3 - 5, consider the function \(f(x)=(1+x)^{1/x}\).

3) 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 \(f(x)=(1+x)^{1/x}\)?

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) \(\displaystyle \lim_{x \to 0}\frac{\sin 2x}{x};\quad \pm 0.1,\; \pm 0.01, \; \pm 0.001, \; \pm .0001\)

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

7) \(\displaystyle \lim_{x \to 0}\frac{\sin 3x}{x} \pm 0.1, \; \pm 0.01, \; \pm 0.001, \; \pm 0.0001\)

\(x\)	\(\frac{\sin 3x}{x}\)	\(x\)	\(\frac{\sin 3x}{x}\)
-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: \(\displaystyle \lim_{x \to 0}\frac{\sin ax}{x}\) for \(a\), a positive real value.

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.

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.

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.

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) \(\displaystyle \lim_{ \theta \to 0}\sin\left(\frac{ \pi }{ \theta }\right)\)

\( \theta \)	\(\sin\left(\frac{ \pi }{ \theta }\right)\)	\( \theta \)	\(\sin\left(\frac{ \pi }{ \theta }\right)\)
-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) \(\displaystyle \lim_{ \alpha \to 0^+} \frac{1}{ \alpha }\cos\left(\frac{ \pi }{ \alpha }\right)\)

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

17) 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 \(\displaystyle \lim_{t \to 2}x(t)\)? 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