Example 4 from Precalculus, 12.1 Finding Limits: Numerical and Graphical Approaches

Using a Graphing Utility to Determine a Limit

With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as $x$ approaches 0. If the function has a limit as $x$ approaches 0, state it. If not, discuss why there is no limit.

f (x) = 3 \sin (\frac{π}{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 use a graphing utility to investigate the behavior of the graph close to $x = 0.$ Centering around $x = 0,$ we choose two viewing windows such that the second one is zoomed in closer to $x = 0$ than the first one. The result would resemble Figure 12 for $[- 2, 2]$ by $[- 3, 3] .$

Figure 12

The result would resemble Figure 13 for $[−0.1, 0.1]$ by $[−3, 3] .$

Figure 13 Even closer to zero, we are even less able to distinguish any limits.

The closer we get to 0, the greater the swings in the output values are. That is not the behavior of a function with either a left-hand limit or a right-hand limit. And if there is no left-hand limit or right-hand limit, there certainly is no limit to the function $f (x)$ as $x$ approaches 0.

We write

$\lim_{x \to 0^{-}} (3 \sin (\frac{π}{x})) does not exist .$

$\lim_{x \to 0^{+}} (3 \sin (\frac{π}{x})) does not exist .$

$\lim_{x \to 0} (3 \sin (\frac{π}{x})) does not exist .$

Your turn

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

Numerically estimate the following limit: $\lim_{x \to 0} (\sin (\frac{2}{x})) .$

How did it go?

Solution

Your turn

Solution

Your turn