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.

limx−>af(x)=L(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.

limx−>a+f(x)=L(2)

limx−>a−f(x)=L(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⎧⎩⎨limx−>a+f(x)=Llimx−>a−f(x)=L⟹limx−>af(x)=L

if⎧⎩⎨limx−>a+f(x)=L1limx−>a−f(x)=L2(L1≠L2)⇏limx−>af(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