Finding Limits: Learn It 4

Finding a Limit Using a Table

Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of [latex]x[/latex] approach [latex]a[/latex] from both sides. Then we determine if the output values get closer and closer to some real value, the limit [latex]L[/latex].

Let’s consider an example using the following function:

[latex]\underset{x\to 5}{\mathrm{lim}}\left(\dfrac{{x}^{3}-125}{x - 5}\right)[/latex]

To create the table, we evaluate the function at values close to [latex]x=5[/latex]. We use some input values less than 5 and some values greater than 5. The table values show that when [latex]x>5[/latex] but nearing 5, the corresponding output gets close to 75. When [latex]x>5[/latex] but nearing 5, the corresponding output also gets close to 75.

Table shows that as x values approach 5 from the positive or negative direction, f(x) gets very close to 75. But when x is equal to 5, y is undefined.

Because

[latex]\underset{x\to {5}^{-}}{\mathrm{lim}}f\left(x\right)=75=\underset{x\to {5}^{+}}{\mathrm{lim}}f\left(x\right)[/latex],

then

[latex]\underset{x\to 5}{\mathrm{lim}}f\left(x\right)=75[/latex].

Remember that [latex]f\left(5\right)[/latex] does not exist.

How To: Given a function [latex]f[/latex], use a table to find the limit as [latex]x[/latex] approaches [latex]a[/latex] and the value of [latex]f\left(a\right)[/latex], if it exists.

  1. Choose several input values that approach [latex]a[/latex] from both the left and right. Record them in a table.
  2. Evaluate the function at each input value. Record them in the table.
  3. Determine if the table values indicate a left-hand limit and a right-hand limit.
  4. If the left-hand and right-hand limits exist and are equal, there is a two-sided limit.
  5. Replace [latex]x[/latex] with [latex]a[/latex] to find the value of [latex]f\left(a\right)[/latex].
Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit.

[latex]\underset{x\to 0}{\mathrm{lim}}\left(\dfrac{5\sin \left(x\right)}{3x}\right)[/latex]

Graph the function on a viewing window. A graphical check shows both branches of the graph of the function get close to the output 75 as [latex]x[/latex] nears 5. Furthermore, we can use the ‘trace’ feature of a graphing calculator. By approaching [latex]x=5[/latex] we may numerically observe the corresponding outputs getting close to [latex]75[/latex].Graph of an increasing function with a discontinuity at (5, 75)

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

[latex]\underset{x\to 0}{\mathrm{lim}}\left(\dfrac{20\sin \left(x\right)}{4x}\right)[/latex]

Both methods have advantages. Graphing allows for quick inspection. Tables can be used when graphical utilities aren’t available, and they can be calculated to a higher precision than could be seen with an unaided eye inspecting a graph.
With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as [latex]x[/latex] approaches 0. If the function has a limit as [latex]x[/latex] approaches 0, state it. If not, discuss why there is no limit.

[latex]f\left(x\right)=3\sin \left(\frac{\pi }{x}\right)[/latex]

Numerically estimate the following limit: [latex]\underset{x\to 0}{\mathrm{lim}}\left(\sin \left(\dfrac{2}{x}\right)\right)[/latex].