 # 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 $a$.

d. The second column also shows that $\boldsymbol{$\lim_{x \rightarrow 3^{+}} f(x)} = 4$. Comparing the results of a and b shows that the regular limit does not exist since the two limits are not equal.$\begin{aligned} \lim_{x \rightarrow 1^{-}} f(x) &= \dfrac{1}{2}\\\lim_{x \rightarrow 1^{+}} f(x) &= 2\sqrt{6}\\\lim_{x \rightarrow 1^{-}} f(x)&\neq \lim_{x \rightarrow 1^{+}} f(x)\end{aligned}$Comparing the results of c and d, we can show that the regular limit exists since the two limits are equal.$\begin{aligned} \lim_{x \rightarrow 3^{-}} f(x) &= 4\\\lim_{x \rightarrow 3^{+}} f(x) &= 4\\\lim_{x \rightarrow 3^{-}} f(x)&= \lim_{x \rightarrow 3^{+}} f(x)\end{aligned}$### Practice Questions 1. The graph of$f(x)$is as shown below. What is the value of$\lim_{x \rightarrow 3^{-}} f(x)$? 2. The graph of$f(x)$is as shown below. What is the value of$\lim_{x \rightarrow 3^{+}} f(x)$? 3. The graph of$f(x)$is as shown below. What is the value of$\lim_{x \rightarrow -6^{-}} f(x)$? 4. The graph of$f(x)$is as shown below. What is the value of$\lim_{x \rightarrow 6^{+}} f(x)$? 5. Using the table of values for$f(x)$as shown below, which of the following statements describe the one-sided limits as$x$approaches$-5$? $\boldsymbol{x}\boldsymbol{f(x)}\boldsymbol{x}\boldsymbol{f(x)}-6-15.01548-4.99991.978421-5.9-15.12548-4.9991.687899-5.5-15.48792-4.991.467821-5.1-15.64870-4.91.324159-5.01-15.80549-4.51.132497-5.001-15.92157-41.021578$6. The function$f(x)$is a piecewise function as shown by the equation below.$ f(x)=\left\{\begin{matrix}\dfrac{2x}{x+2},& 0 \leq x < 2\\ \sqrt{16-x^2},&2 \leq x \leq 4\\3x + 2,&4 < x \leq 10\end{matrix}\right.$What is the value of$\lim_{x \rightarrow 2^{-}} f(x)$? 7. The function$f(x)$is a piecewise function as shown by the equation below.$ f(x)=\left\{\begin{matrix}\dfrac{2x}{x+2},& 0 \leq x < 2\\ \sqrt{16-x^2},&2 \leq x \leq 4\\3x + 2,&4 < x \leq 10\end{matrix}\right.$What is the value of$\lim_{x \rightarrow 2^{+}} f(x)$? 8. The function$f(x)$is a piecewise function as shown by the equation below.$ f(x)=\left\{\begin{matrix}\dfrac{2x}{x+2},& 0 \leq x < 2\\ \sqrt{16-x^2},&2 \leq x \leq 4\\3x + 2,&4 < x \leq 10\end{matrix}\right.$What is the value of$\lim_{x \rightarrow 3^{-}} f(x)$? 9. The function$f(x)$is a piecewise function as shown by the equation below.$ f(x)=\left\{\begin{matrix}\dfrac{2x}{x+2},& 0 \leq x < 2\\ \sqrt{16-x^2},&2 \leq x \leq 4\\3x + 2,&4 < x \leq 10\end{matrix}\right.$What is the value of$\lim_{x \rightarrow 3^{+}} f(x)\$?

Images/mathematical drawings are created with GeoGebra.

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