Introduction to the Limit of a Function: Learn It 3

One-Sided Limits

Sometimes indicating that the limit of a function fails to exist at a point does not provide us with enough information about the behavior of the function at that particular point.

To see this, we now revisit the function [latex]g(x)=\frac{|x-2|}{(x-2)}[/latex] introduced at the beginning of the section (Figure 1 part b).

"Three 2 and x= -1 for x < 2. There are open circles at both endpoints (2, 1) and (-2, 1). The third is h(x) = 1 / (x-2)^2, in which the function curves asymptotically towards y=0 and x=2 in quadrants one and two." width="975" height="434"
Figure 1. These graphs show the behavior of three different functions around [latex]x=2[/latex].

As we pick values of [latex]x[/latex] close to [latex]2[/latex], [latex]g(x)[/latex] does not approach a single value, so the limit as [latex]x[/latex] approaches [latex]2[/latex] does not exist—that is, [latex]\underset{x\to 2}{\lim}g(x)[/latex] DNE.

However, this statement alone does not give us a complete picture of the behavior of the function around the [latex]x[/latex]-value [latex]2[/latex]. To provide a more accurate description, we introduce the idea of a one-sided limit.

For all values to the left of [latex]2[/latex] (or the negative side of [latex]2[/latex]), [latex]g(x)=-1[/latex]. Thus, as [latex]x[/latex] approaches [latex]2[/latex] from the left, [latex]g(x)[/latex] approaches [latex]−1[/latex].

Mathematically, we say that the limit as [latex]x[/latex] approaches [latex]2[/latex] from the left is [latex]−1[/latex]. Symbolically, we express this idea as

[latex]\underset{x\to 2^-}{\lim}g(x)=-1[/latex]

Similarly, as [latex]x[/latex] approaches [latex]2[/latex] from the right (or from the positive side), [latex]g(x)[/latex] approaches [latex]1[/latex]. Symbolically, we express this idea as

[latex]\underset{x\to 2^+}{\lim}g(x)=1[/latex]

one-sided limits

One-sided limits are limits approached from one direction—either from the left or the right.

  • Left-Sided Limit: For a function [latex]f(x)[/latex] on an interval ending at [latex]a[/latex], if [latex]f(x)[/latex] approaches a specific value [latex]L[/latex] as the values of [latex]x[/latex] approaches [latex]a[/latex] from the left ([latex]x < a[/latex]), we denote this limit as:
    [latex]\underset{x\to a^-}{\lim}f(x)=L[/latex]
  • Right-Sided Limit: For a function [latex]f(x)[/latex] on an interval ending at [latex]a[/latex], if [latex]f(x)[/latex] approaches a specific value [latex]L[/latex] as the values of [latex]x[/latex] approaches [latex]a[/latex] from the right  ([latex]x > a[/latex]), we express this limit as:
    [latex]\underset{x\to a^+}{\lim}f(x)=L[/latex]

For the function [latex]f(x)=\begin{cases} x+1, & \text{ if } \, x < 2 \\ x^2-4, & \text{ if } \, x \ge 2 \end{cases}[/latex], evaluate each of the following limits.

  1. [latex]\underset{x\to 2^-}{\lim}f(x)[/latex]
  2. [latex]\underset{x\to 2^+}{\lim}f(x)[/latex]

Two-Sided Limits

To fully grasp how limits function, it’s essential to understand the connection between one-sided and two-sided limits.

A two-sided limit at a point exists only if the one-sided limits from both the left and the right converge to the same value. If there’s a discrepancy between the left and the right limits, the two-sided limit at that point does not exist.

two-sided limits

For a function [latex]f(x)[/latex], defined over an interval including [latex]a[/latex] (except possibly at [latex]a[/latex] itself), we say the two-sided limit exists as [latex]x[/latex] approaches [latex]a[/latex] and equals [latex]L[/latex] if, and only if, both one-sided limits as [latex]x[/latex] approaches [latex]a[/latex] also equals [latex]L[/latex].

[latex]\underset{x\to a}{\lim}f(x)=L[/latex], if and only if [latex]\underset{x\to a^-}{\lim}f(x)=L[/latex] and [latex]\underset{x\to a^+}{\lim}f(x)=L[/latex]