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 $$\displaystyle \lim_ f(x)$$.

$$ \begin & \\ \hline 4.5 & 8.32571\\\hline 4.75 & 8.95692\\\hline 4.9 & 8.99084\\\hline 4.99 & 8.99987\\\hline 4.999 & 8.99992\\\hline 4.9999 & 8.99999\\\hline \end $$

$$ \begin & \\ \hline 5.5 & 9.64529\\\hline 5.25 & 9.26566\\\hline 5.1 & 9.04215\\\hline 5.01 & 9.00113\\\hline 5.001 & 9.00011\\\hline 5.0001 & 9.00001\\\hline \end $$

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

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

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

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

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

Answer: $$\displaystyle \lim_ f(x) \approx 9$$.

Example 2: Using Tables to Estimate Limits

Using the tables below, estimate $$\displaystyle \lim_ f(x)$$.

$$ \begin & \\ \hline -8.5 & 13.1365\\\hline -8.1 & -2.4336\\\hline -8.01 & -2.91313\\\hline -8.001 & -2.99131\\\hline -8.0001 & -2.99913\\\hline -8.00001 & -2.99991\\\hline \end $$

$$ \begin & \\ \hline -7.5 & -6\\\hline -7.9 & -5.5\\\hline -7.99 & -5.15\\\hline -7.999 & -5.015\\\hline -7.9999 & -5.0015\\\hline -7.99999 & -5.00015\\\hline \end $$

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

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

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

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

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

Answer: $$\displaystyle \lim_ f(x)$$ does not exist.