12.1 Finding limits: numerical and graphical approaches  (Page 5/10)

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.

$$f(x)=3\sin \left(\frac{\pi }{x}\right)$$

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 [link] for $[-2,2]$ by $[-3,3].$

The result would resemble [link] for $[\mathrm{-0.1},0.1]$ by $[\mathrm{-3},3].$

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 $f\left(x\right)$ as $x$ approaches 0.

We write

$$\underset{x\to 0^{-}}{\lim }\left(3\sin \left(\frac{\pi }{x}\right)\right)\text{ does not exist}.$$

$$\underset{x\to 0^{+}}{\lim }\left(3\sin \left(\frac{\pi }{x}\right)\right)\text{ does not exist}.$$

$$\underset{x\to 0}{\lim }\left(3\sin \left(\frac{\pi }{x}\right)\right)\text{ does not exist}\text{.}$$

Got questions? Get instant answers now!

Got questions? Get instant answers now!

Key concepts

• A function has a limit if the output values approach some value $L$ as the input values approach some quantity $a.$ See [link] .
• A shorthand notation is used to describe the limit of a function according to the form $\underset{x\to a}{\lim }f(x)=L,$ which indicates that as $x$ approaches $a,$ both from the left of $x=a$ and the right of $x=a,$ the output value gets close to $L.$
• A function has a left-hand limit if $f\left(x\right)$ approaches $L$ as $x$ approaches $a$ where $x<a.$ A function has a right-hand limit if $f\left(x\right)$ approaches $L$ as $x$ approaches $a$ where $x>a.$
• A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.
• A graph provides a visual method of determining the limit of a function.
• If the function has a limit as $x$ approaches $a,$ the branches of the graph will approach the same $y\text{-}$ coordinate near $x=a$ from the left and the right. See [link] .
• A table can be used to determine if a function has a limit. The table should show input values that approach $a$ from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit. See [link] .
• A graphing utility can also be used to find a limit. See [link] .

Section exercises

Verbal