Estimating Limit Values with Tables

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When working with tables, the best we can do is estimate the limit value.

Examples

Example 1: Using Tables to Estimate Limits

Use the tables shown below to estimate the value of $$\displaystyle \lim_{x\to 5} f(x)$$.

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 4.5 & 8.32571\\\hline 4.75 & 8.95692\\\hline 4.9 & 8.99084\\\hline 4.99 & 8.99987\\\hline 4.999 & 8.99992\\\hline 4.9999 & 8.99999\\\hline \end{array} $$

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 5.5 & 9.64529\\\hline 5.25 & 9.26566\\\hline 5.1 & 9.04215\\\hline 5.01 & 9.00113\\\hline 5.001 & 9.00011\\\hline 5.0001 & 9.00001\\\hline \end{array} $$

Step 1

Examine what happens as $$x$$ approaches from the left.

As the $$x$$-values approach 5...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 4.5 & 8.32571\\\hline 4.75 & 8.95692\\\hline 4.9 & 8.99084\\\hline 4.99 & 8.99987\\\hline 4.999 & 8.99992\\\hline 4.9999 & 8.99999\\\hline \end{array} $$
...$$f(x)$$ seems to approach 9.

Step 2

Examine what happens as $$x$$ approaches from the right.

As the $$x$$-values approach 5...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 5.5 & 9.64529\\\hline 5.25 & 9.26566\\\hline 5.1 & 9.04215\\\hline 5.01 & 9.00113\\\hline 5.001 & 9.00011\\\hline 5.0001 & 9.00001\\\hline \end{array} $$
...$$f(x)$$ seems to approach 9.

Step 3

If the function seems to approach the same value from both directions, then that is the estimate of the limit value.

Answer: $$\displaystyle \lim_{x\to 5} f(x) \approx 9$$.

Example 2: Using Tables to Estimate Limits

Using the tables below, estimate $$\displaystyle \lim_{x\to-8} f(x)$$.

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline -8.5 & 13.1365\\\hline -8.1 & -2.4336\\\hline -8.01 & -2.91313\\\hline -8.001 & -2.99131\\\hline -8.0001 & -2.99913\\\hline -8.00001 & -2.99991\\\hline \end{array} $$

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline -7.5 & -6\\\hline -7.9 & -5.5\\\hline -7.99 & -5.15\\\hline -7.999 & -5.015\\\hline -7.9999 & -5.0015\\\hline -7.99999 & -5.00015\\\hline \end{array} $$

Step 1

Examine what happens as $$x$$ approaches from the left.

As the $$x$$-values approach -8...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline -8.5 & 13.1365\\\hline -8.1 & -2.4336\\\hline -8.01 & -2.91313\\\hline -8.001 & -2.99131\\\hline -8.0001 & -2.99913\\\hline -8.00001 & -2.99991\\\hline \end{array} $$
...$$f(x)$$ seems to approach -3.

Step 2

Examine what happens as $$x$$ approaches from the right.

As the $$x$$-values approach -8...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline -7.5 & -4\\\hline -7.9 & -3.5\\\hline -7.99 & -3.15\\\hline -7.999 & -3.015\\\hline -7.9999 & -3.0015\\\hline -7.99999 & -3.00015\\\hline \end{array} $$
...$$f(x)$$ seems to approach -3.

Step 3

If the function seems to approach different values, then the limit does not exist.

Answer: $$\displaystyle \lim_{x\to-8} f(x)$$ does not exist.

Practice Problems

Problem 1

Estimate $$\lim\limits_{x\to12} f(x)$$ using the tables below.

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 11.5 & 7\\\hline 11.9 & 7.5\\\hline 11.99 & 7.9\\\hline 11.999 & 7.99\\\hline 11.9999 & 7.999\\\hline 11.99999 & 7.9999\\\hline \end{array} $$

$$ \begin{array}{l|c} {x} & f(x)\\ \hline 12.5 & 8.5\\\hline 12.1 & 8.1\\\hline 12.01 & 8.01\\\hline 12.001 & 8.001\\\hline 12.0001 & 8.0001\\\hline 12.00001 & 8.00001\\\hline \end{array} $$

Step 1

Examine what happens as $$x$$ approaches from the left.

As the $$x$$-values approach 12 from the left...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 11.5 & 7\\\hline 11.9 & 7.5\\\hline 11.99 & 7.9\\\hline 11.999 & 7.99\\\hline 11.9999 & 7.999\\\hline 11.99999 & 7.9999\\\hline \end{array} $$
...$$f(x)$$ seems to approach 8.

Step 2

Examine what happens as $$x$$ approaches from the right.

As the $$x$$-values approach 12 from the right
$$ \begin{array}{l|c} {x} & {f(x))}\\ \hline 12.5 & 8.5\\\hline 12.1 & 8.1\\\hline 12.01 & 8.01\\\hline 12.001 & 8.001\\\hline 12.0001 & 8.0001\\\hline 12.00001 & 8.00001\\\hline \end{array} $$
...f(x) seems to approach 8.

Step 3

If the function seems to approach the same value from both directions, then that is the estimate of the limit value.

$$\displaystyle \lim_{x\to 12} f(x) \approx 8$$.

Problem 2

What can be determined about $$\lim\limits_{x\to\frac{1}{2}} f(x)$$ based on the tables below?/

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 0 & 1.7\\\hline 0.2 & 1.75\\\hline 0.4 & 1.795\\\hline 0.45 & 1.7995\\\hline 0.49 & 1.79995\\\hline 0.499 & 1.79999\\\hline \end{array} $$

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 1 & -2.44445\\\hline 0.8 & -2.55556\\\hline 0.6 & -2.66667\\\hline 0.55 & -2.77778\\\hline 0.51 & -2.88889\\\hline 0.501 & -2.99999\\\hline \end{array} $$

Step 1

Examine what happens as $$x$$ approaches from the left.

As the $$x$$-values approach $$\frac{1}{2}$$ from the left...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 0 & 1.7\\\hline 0.2 & 1.75\\\hline 0.4 & 1.795\\\hline 0.45 & 1.7995\\\hline 0.49 & 1.79995\\\hline 0.499 & 1.79999\\\hline \end{array} $$
...$$f(x)$$ seems to approach 1.8.

Step 2

Examine what happens as $$x$$ approaches from the right.

As the $$x$$-values approach $$\frac{1}{2}$$ from the right
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 1 & -2.44445\\\hline 0.8 & -2.55556\\\hline 0.6 & -2.66667\\\hline 0.55 & -2.77778\\\hline 0.51 & -2.88889\\\hline 0.501 & -2.99999\\\hline \end{array} $$
$$f(x)$$ seems to approach -3.

Step 3

If the function seems to approach different values, then the limit does not exist.

$$\lim\limits_{x\to\frac{1}{2}} f(x)$$ does not exist.

Problem 3

Estimate the value of $$\lim\limits_{x\to0.75} f(x)$$ using the tables below.

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 0.7 & 0.1\\\hline 0.72 & -0.01\\\hline 0.74 & 0.001\\\hline 0.749 & -0.0001\\\hline 0.7499 & 0.00001\\\hline 0.74999 & -0.000001\\\hline \end{array} $$

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 0.8 & 0.3\\\hline 0.78 & -0.06\\\hline 0.76 & 0.0012\\\hline 0.751 & -0.0006\\\hline 0.7501 & 0.00003\\\hline 0.75001 & -0.000006\\\hline \end{array} $$

Step 1

Examine what happens as $$x$$ approaches from the left.

As the $$x$$-values approach 0.75 from the left...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 0.7 & 0.1\\\hline 0.72 & -0.01\\\hline 0.74 & 0.001\\\hline 0.749 & -0.0001\\\hline 0.7499 & 0.00001\\\hline 0.74999 & -0.000001\\\hline \end{array} $$
...$$f(x)$$ seems to approach 0.

Step 2

Examine what happens as $$x$$ approaches from the right.

As the $$x$$-values approach 0.75 from the right...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 0.8 & 0.3\\\hline 0.78 & -0.06\\\hline 0.76 & 0.0012\\\hline 0.751 & -0.0006\\\hline 0.7501 & 0.00003\\\hline 0.75001 & -0.000006\\\hline \end{array} $$
...$$f(x)$$ seems to approach 0.

Step 3

$$\lim\limits_{t\to0.75} f(x) \approx 0$$

Problem 4

Use the tables below to estimate the value of $$\lim\limits_{x\to 10} f(x)$$.

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 9.5 & 2.3\\\hline 9.9 & 1.8\\\hline 9.99 & 8.3\\\hline 9.999 & 0.8\\\hline 9.9999 & 9.8\\\hline 9.99999 & 2.6\\\hline \end{array} $$

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 10.5 & 1.1\\\hline 10.1 & 5.8\\\hline 10.01 & 3.6\\\hline 10.001 & 2.9\\\hline 10.0001 & 5.4\\\hline 10.00001 & 12.5\\\hline \end{array} $$

Step 1

Examine what happens as $$x$$ approaches from the left.

As the $$x$$-values approach 10 from the left...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 9.5 & 2.3\\\hline 9.9 & 1.8\\\hline 9.99 & 8.3\\\hline 9.999 & 0.8\\\hline 9.9999 & 9.8\\\hline 9.99999 & 2.6\\\hline \end{array}; $$
...$$f(x)$$ doesn't seem to approach anything.

Step 2

Examine what happens as $$x$$ approaches from the right

As the $$x$$-values approach 10 from the right...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline 10.5 & 1.1\\\hline 10.1 & 5.8\\\hline 10.01 & 3.6\\\hline 10.001 & 2.9\\\hline 10.0001 & 5.4\\\hline 10.00001 & 12.5\\\hline \end{array}; $$
...$$f(x)$$ doesn’t seem to approach anything.

Step 3

The function doesn't seem to approach a particular value, so the limit does not exist.

$$\lim\limits_{x\to10} f(x)$$ doesn't exist.

Problem 5

Use the tables below to determine $$\lim\limits_{x\to -3} f(x)$$.

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline -4 & 6\\\hline -3.5 & 61\\\hline -3.1 & 611\\\hline -3.01 & 6111\\\hline -3.001 & 61,111\\\hline -3.0001 & 611,111\\\hline \end{array} $$

$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline -2 & 7\\\hline -2.5 & 72\\\hline -2.9 & 788\\\hline -2.99 & 7656\\\hline -2.999 & 77,701\\\hline -2.9999 & 711,000\\\hline \end{array} $$

Step 1

Examine what happens as $$x$$ approaches from the left.

As the $$x$$-values approach -3 from the left...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline -4 & 6\\\hline -3.5 & 61\\\hline -3.1 & 611\\\hline -3.01 & 6111\\\hline -3.001 & 61,111\\\hline -3.0001 & 611,111\\\hline \end{array} $$
...$$f(x)$$ keeps getting larger.

Step 2

Examine what happens as $$x$$ approaches from the right.

As the $$x$$-values approach 3 from the right...
$$ \begin{array}{l|c} {x} & {f(x)}\\ \hline -2 & 7\\\hline -2.5 & 72\\\hline -2.9 & 788\\\hline -2.99 & 7656\\\hline -2.999 & 77,701\\\hline -2.9999 & 711,000\\hline \end{array} $$
...$$f(x)$$ keeps getting larger.

Step 3

The function doesn't seem to approach a particular value, so the limit does not exist.

$$\lim\limits_{x\to-3} f(x)$$ doesn't exist.

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