Example 3: Overall Limit Exists

Determine $\underset{x\to 2}{\lim }f\left(x\right)=$

Step 1: Draw a “wall” at the $a–\mathit{value}$ ($x–\mathit{value}$) you are heading towards.

$\underset{x\to 2}{\lim }f\left(x\right)=$

In this example we will draw the “wall” at $x=2$ because that is the $a–\mathit{value}$ we are heading towards.

Step 2: Determine what type of limit you are being asked to evaluate. LHL ($\mathit{–}$), RHL ($+$), or Overall .

In this example we do not have anything in the exponent of our $a–\mathit{value}$. So, we are being asked to find the Overall Limit.

$\underset{x\to 2}{\lim }f\left(x\right)=$

Step 3 ( Overall ): To determine the overall limit, we must determine both the left and right-hand limits.

LHL : Drawing along your graph, $f\left(x\right)$, starting on the left side of the graph and moving to the right until you run into the “wall” you drew in Step 1 . In this example we run into the “wall” at $y=–1$.

$\underset{x\to 2^{–}}{\lim }f\left(x\right)=–1$

RHL : Begin drawing along your graph, $f\left(x\right)$, starting on the right side of the graph and moving to the left until you run into the “wall” you drew in Step 1 . In this example we run into the “wall” at $y=–1$.

$\underset{x\to 2^{+}}{\lim }f\left(x\right)=–1$

Overall : We now compare the LHL and the RHL to determine the final result for the overall limit.

Final Result:

In this example the LHL and the RHL both have results $y=–1$.

Left-Hand Limit = Right-Hand Limit = $–1$

$\underset{x\to 2^{–}}{\lim }f\left(x\right)=\underset{x\to 2^{\mathbf{+}}}{\lim }f\left(x\right)=–1$

Overall Limit =$\underset{x\to 2}{\lim }f\left(x\right)=\mathrm{ }–1$