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Finding a Limit Using a Table · Physics Playground

Exercises Limits and derivatives

Example 3 from Precalculus, 12.1 Finding Limits: Numerical and Graphical Approaches

Finding a Limit Using a Table

Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit.

\lim_{x \to 0} (\frac{5 \sin (x)}{3 x})

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We can estimate the value of a limit, if it exists, by evaluating the function at values near $x = 0.$ We cannot find a function value for $x = 0$ directly because the result would have a denominator equal to 0, and thus would be undefined.
$f (x) = \frac{5 \sin (x)}{3 x}$

We create Figure 10 by choosing several input values close to $x = 0,$ with half of them less than $x = 0$ and half of them greater than $x = 0.$ Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table.
The table values indicate that when $x < 0$ but approaching 0, the corresponding output nears $\frac{5}{3} .$

When $x > 0$ but approaching 0, the corresponding output also nears $\frac{5}{3} .$

Figure 10

Because

$\lim_{x \to 0^{-}} f (x) = \frac{5}{3} = \lim_{x \to 0^{+}} f (x),$

then

$\lim_{x \to 0} f (x) = \frac{5}{3} .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Numerically estimate the limit of the following function by making a table:

\lim_{x \to 0} (\frac{20 \sin (x)}{4 x})

How did it go?

PreviousFinding a Limit Using a Graph Next Using a Graphing Utility to Determine a Limit

From Precalculus (1st edition) by OpenStax (Rice University), section 12.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.

Solution

Your turn