Evaluating Limits from Tables

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) $2.9, 2.99, 2.999, 3.001, 3.01, 3.1$
Answer: $2.9, 2.99, 2.999, 3.001, 3.01, 3.1$
A good table approaches $3$ from both sides and gets closer each time: $2.9, 2.99, 2.999$ from the left and $3.001, 3.01, 3.1$ from the right. Values like $1, 2, 4, 5$ are too far away, one side alone can miss a jump, and $x = 3$ itself should not be used.
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 $\lim_{x \to 5} f(x)$ .
Answer: 10
From the left the values $9.9$ , $9.99$ , $9.999$ climb toward $10$ ; from the right $10.1$ , $10.01$ , $10.001$ fall toward $10$ . Both sides agree, so $\lim_{x \to 5} f(x) = 10$ .
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) $6.998$
Answer: $7$
The left values $6.8$ , $6.98$ , $6.998$ and the right values $7.2$ , $7.02$ , $7.002$ both close in on $7$ . The entries $6.998$ and $7.002$ are only estimates; the limit is $7$ .
4.
For $f(x) = \frac{x^2 - 9}{x - 3}$ , compute the table entry $f(2.99)$ .
Answer: 5.99
$f(2.99) = \frac{8.9401 - 9}{2.99 - 3} = \frac{-0.0599}{-0.01} = 5.99$ . The table values head toward $6$ as $x \to 3$ .
5.
Use a table to estimate $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$ .
Answer: 2
1. Compute $f(0.9)$ . 1.9
2. Compute $f(0.99)$ . 1.99
3. Compute $f(1.01)$ . 2.01
4. Which number do both sides approach? 2
6.
Use a table to estimate $\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$ .
Answer: 8
1. Compute $f(3.9)$ . 7.9
2. Compute $f(3.99)$ . 7.99
3. Compute $f(4.01)$ . 8.01
4. Which number do both sides approach? 8
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 $-2$
Answer: The limit does not exist
From the left $g(x)$ stays at $-2$ , from the right it stays at $5$ . The one-sided values disagree ( $-2 \neq 5$ ), so the two-sided limit does not exist. Averaging them ( $1.5$ ) is not allowed.
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 $\lim_{x \to 0} f(x)$ .
Answer: 1
From the left: $0.99502$ , $0.99950$ rise toward $1$ . From the right: $1.00502$ , $1.00050$ fall toward $1$ . So $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$ . This function cannot be factored, which is exactly when a table is useful.
9.
Use a table to estimate $\lim_{x \to 0} \frac{\sin x}{x}$ , with $x$ in radians.
Answer: 1
1. Compute $f(0.1)$ , rounded to four decimal places. 0.9983
2. Compute $f(0.01)$ , rounded to four decimal places. 1.0000
3. Compute $f(-0.1)$ , rounded to four decimal places. 0.9983
4. Which number do both sides approach? 1
10.
Use a table to estimate $\lim_{x \to 2} \frac{x^3 - 8}{x - 2}$ .
Answer: 12
1. Compute $f(1.9)$ . 11.41
2. Compute $f(1.99)$ . 11.9401
3. Compute $f(2.01)$ . 12.0601
4. Which number do both sides approach? 12
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.
Answer: 0.167
Both sides close in on about $0.1667$ , which rounds to $0.167$ . Algebra confirms it: the exact limit is $\frac{1}{6} \approx 0.16667$ .
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
Answer: $h(x)$ grows without bound, so there is no finite limit
Each step closer to $1$ makes $h(x)$ $100$ times bigger, from both sides. The values grow without bound, so there is no finite limit; we write $\lim_{x \to 1} h(x) = \infty$ . The value $1\,000\,000$ is just the last entry, not a 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.
Answer: The limit does not exist: the left side approaches $3$ , the right side approaches $4$
1. Compute $f(1.99)$ . 2.99
2. Compute $f(2.01)$ . 4.0401
3. Which number does the left side approach? 3
4. Which number does the right side approach? 4
5. Does $\lim_{x \to 2} f(x)$ exist? no
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.
Answer: 2.72
The closest values $2.71964$ and $2.71692$ squeeze the limit between them, and both round to $2.72$ . The exact limit is the number $e \approx 2.71828$ .
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
Answer: The limit does not exist
The chosen values $x = \pm 0.1, \pm 0.01, \pm 0.001$ make $\frac{\pi}{x}$ a multiple of $\pi$ , so they all hit zeros. Arbitrarily close to $0$ there are also $x$ values giving $1$ and $-1$ , so $f(x)$ keeps oscillating and the limit does not exist. A table alone was misleading here.
16.
Use a table to estimate $\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$ .
Answer: 0.25
1. Compute $f(3.9)$ , rounded to four decimal places. 0.2516
2. Compute $f(3.99)$ , rounded to four decimal places. 0.2502
3. Compute $f(4.01)$ , rounded to four decimal places. 0.2498
4. Which decimal number do both sides approach? 0.25
17.
Use a table to describe $\lim_{x \to -1} \frac{1}{(x + 1)^2}$ .
Answer: $\infty$
1. Compute $f(-1.1)$ . 100
2. Compute $f(-1.01)$ . 10000
3. Compute $f(-0.99)$ . 10000
4. How many times bigger is $f(-1.01)$ than $f(-1.1)$ ? 100
18.
Investigate $\lim_{x \to 0} \sin\left(\frac{\pi}{x}\right)$ with carefully chosen table values ( $x$ in radians).
Answer: The limit does not exist: near $0$ the values keep swinging between $-1$ and $1$
1. Compute $f(0.1)$ . 0
2. Compute $f(0.4)$ . 1
3. Compute $f\left(\frac{2}{7}\right)$ . -1
4. Does $\lim_{x \to 0} \sin\left(\frac{\pi}{x}\right)$ exist? no