# One sided limits – Definition, Techniques, and Examples

What happens when the graph of a function contains discontinuities such as piecewise and rational functions? The function’s behavior may differ as the function approaches from the left or the right, so we instead inspect its one-sided limits.

One-sided limits are the function’s limits as they approach a restricted value or side of the function.

This article will review discontinuities and how they affect the graph’s limit as it approaches from the left or right of $x = a$. We’ll also introduce you to the techniques of finding the limits of a function from the left and from the right.

## What is a one sided limit?

A one-sided limit can either be of the form $\lim_{x \rightarrow a^{-}} f(x)$ or $\lim_{x \rightarrow a^{+}} f(x)$. The plus or minus sign following $a$ indicate the region or interval that we want to observe.

From the notations alone, we can see that one-sided limits are the limits of the function as it approaches a certain value and interval.

One-sided limits are helpful when we want to check for discontinuities and if we want to confirm if a limit exists. Let’s observe the piecewise function, $f(x)=\left\{\begin{matrix}x,&x < 0\\ 5 – x,&x \geq 0\end{matrix}\right.$, and the behavior of $x$ as it approaches $0$.

This piecewise function is a good example of why we study one-sided limits. As $x$ approaches $0$ from the left, we can see that the limit of $f(x)$ is equal to $0$, and when $x$ approaches $0$ from the right, the limit becomes $5$.

This shows that it is possible for the limit from each side of the function may be different.

### What does the limit from the left mean?

Let’s dive into understanding what $\lim_{x \rightarrow a^{-}} f(x)$ represent. When we see $\lim_{x \rightarrow a^{-}}$ , this means that the limit is evaluated from the left of $a$.

When we say limit from the left of $a$, we want to find the limit of $f(x)$ as $x<a$ without letting $x$ be $a$.

For the piecewise function, $f(x)=\left\{\begin{matrix}x,&x < 0\\ 5 – x,&x \geq 0\end{matrix}\right.$, $\lim_{x \rightarrow 0^{-} }f(x)$ represents the limit of the function as $x$ approaches zero coming from the values that are less than 0.

### What does the limit from the right mean?

Similarly, whenever we see $\lim_{x\rightarrow a^{+} }$, this means that the limit is evaluated from the right of $a$ or when $x<a$ without letting $x$ be $a$.

Evaluate the