Evaluating Limits from Tables

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Show your work for each problem.

1.
You want to estimate $\lim_{x \to 3} f(x)$ with a table of values. Which set of $x$ values is the best choice?
1. a) $3, 3.1, 3.2, 3.3$
2. b) $1, 2, 4, 5$
3. c)only $2.9, 2.99, 2.999$
4. d) $2.9, 2.99, 2.999, 3.001, 3.01, 3.1$
2.
A table gives $f(4.9) = 9.9$ , $f(4.99) = 9.99$ , $f(4.999) = 9.999$ , $f(5.001) = 10.001$ , $f(5.01) = 10.01$ , $f(5.1) = 10.1$ . Estimate $\lim_{x \to 5} f(x)$ .
3.
Left of $2$ : $f(1.9) = 6.8$ , $f(1.99) = 6.98$ , $f(1.999) = 6.998$ . Right of $2$ : $f(2.1) = 7.2$ , $f(2.01) = 7.02$ , $f(2.001) = 7.002$ . What is $\lim_{x \to 2} f(x)$ ?
1. a)does not exist
2. b) $7$
3. c) $7.002$
4. d) $6.998$
4.
For $f(x) = \frac{x^2 - 9}{x - 3}$ , compute the table entry $f(2.99)$ .
5.
Use a table to estimate $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$ .
6.
Use a table to estimate $\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$ .
7.
A table for $g(x)$ near $x = 1$ shows $g(0.9) = -2$ , $g(0.99) = -2$ , $g(0.999) = -2$ , $g(1.001) = 5$ , $g(1.01) = 5$ , $g(1.1) = 5$ . What can you conclude about $\lim_{x \to 1} g(x)$ ?
1. a)The limit is $1.5$
2. b)The limit does not exist
3. c)The limit is $5$
4. d)The limit is $-2$
8.
A table for $f(x) = \frac{e^x - 1}{x}$ gives $x = -0.01$ : $0.99502$ , $x = -0.001$ : $0.99950$ , $x = 0.001$ : $1.00050$ , $x = 0.01$ : $1.00502$ . Estimate $\lim_{x \to 0} f(x)$ .
9.
Use a table to estimate $\lim_{x \to 0} \frac{\sin x}{x}$ , with $x$ in radians.
10.
Use a table to estimate $\lim_{x \to 2} \frac{x^3 - 8}{x - 2}$ .
11.
For $f(x) = \frac{\sqrt{x + 9} - 3}{x}$ a table gives $f(-0.01) \approx 0.16671$ , $f(-0.001) \approx 0.16667$ , $f(0.001) \approx 0.16666$ , $f(0.01) \approx 0.16662$ . Estimate $\lim_{x \to 0} f(x)$ as a decimal, rounded to three decimal places.
12.
A table for $h(x)$ near $x = 1$ shows $h(0.9) = 100$ , $h(0.99) = 10\,000$ , $h(0.999) = 1\,000\,000$ , $h(1.001) = 1\,000\,000$ , $h(1.01) = 10\,000$ , $h(1.1) = 100$ . Which statement best describes $\lim_{x \to 1} h(x)$ ?
1. a)The limit is $100$
2. b)The limit is $0$
3. c)The limit is $1\,000\,000$
4. d) $h(x)$ grows without bound, so there is no finite limit
13.
Let $f(x) = x + 1$ for $x < 2$ and $f(x) = x^2$ for $x \geq 2$ . Use a table to decide whether $\lim_{x \to 2} f(x)$ exists.
14.
A table for $f(x) = (1 + x)^{\frac{1}{x}}$ gives $x = -0.01$ : $2.73200$ , $x = -0.001$ : $2.71964$ , $x = 0.001$ : $2.71692$ , $x = 0.01$ : $2.70481$ . Estimate $\lim_{x \to 0} f(x)$ , rounded to two decimal places.
15.
A table for $f(x) = \sin\left(\frac{\pi}{x}\right)$ with $x = \pm 0.1, \pm 0.01, \pm 0.001$ shows $0$ every time. But $f\left(\frac{2}{101}\right) = 1$ and $f\left(\frac{2}{103}\right) = -1$ , and both of these $x$ values are within $0.02$ of $0$ . What is $\lim_{x \to 0} f(x)$ ?
1. a) $1$
2. b) $0$
3. c) $\infty$
4. d)The limit does not exist
16.
Use a table to estimate $\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$ .
17.
Use a table to describe $\lim_{x \to -1} \frac{1}{(x + 1)^2}$ .
18.
Investigate $\lim_{x \to 0} \sin\left(\frac{\pi}{x}\right)$ with carefully chosen table values ( $x$ in radians).