What is a one-sided limits?

What is a one-sided limits?

Introduction

One-sided Limits and Two-sided Limits

People are familiar with two sided limits, shown below.

$$\lim\limits_{x->a} f(x) = L \tag{$1$}$$

But here, we are going to introduce one-sided limits and shown the correlative between them.
The expression of one-sided limits are shown below.

$$\lim\limits_{x->a^+} f(x) = L \tag{$2$}$$
$$\lim\limits_{x->a^-} f(x) = L \tag{$3$}$$

The different is the superscript of a, which means the direction of approaching. E.g., the formula (2) means that $f(x)$ is close to $L$, as the provided $x$ is approach $a$ from the right hand side.

And the same as formula (3), which means that $f(x)$ is close to $L$, as the provided $x$ is approach $a$ from the left hand side.

Graph

See the graph down there.

1. one-sided limits

Compare graph with formula (2) and (3). You can say that $f(x)$ is getting close to 1 if $x$ approach 2 from the left-hand side. And you can also say that $f(x)$ is getting close to 3 if $x$ approach 2 from the right-hand side.

Correlation between One-sided Limits and Two-sided Limits

When one-sided limits from both sides are equal to the same value, we could simply use two-sided limits to express. Whereas, the one-sided limits is not exactly the same, we have use one-sided limits.

E.g.,

$$if \left\{ \begin{align} \lim\limits_{x->a^+} f(x) = L \\ \lim\limits_{x->a^-} f(x) = L \end{align} \right. \Longrightarrow \lim\limits_{x->a} f(x) = L$$

$$if \left\{ \begin{align} \lim\limits_{x->a^+} f(x) = L_1 \\ \lim\limits_{x->a^-} f(x) = L_2 \end{align} \right. \text{($L_1 \neq L_2$)} \nRightarrow \lim\limits_{x->a} f(x) = L$$

Summary

1. One-sided limits is not equal to two-sided limits.
2. If both sides of one-sided limits have the same value, we could make it two-sided limits.

Reference

[1] Introduction of Calculus in Coursera by Jim Fowler