Example 4: Overall Limit Does Not Exist

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

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

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

In this example we will draw the “wall” at $x=–1$ 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.

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

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 $y=4$.

$\underset{x\to \mathrm{ }{–1}^{–}}{\lim }f\left(x\right)=4$

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 $y=–3$.

$\underset{x\to {–1}^{+}}{\lim }f\left(x\right)=–3$

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 different results.

Left-Hand Limit = $\underset{x\to \mathrm{ }{–1}^{–}}{\lim }f\left(x\right)=4$

Right-Hand Limit = $\underset{x\to {–1}^{+}}{\lim }f\left(x\right)=–3$

Left-Hand Limit ≠ Right-Hand Limit

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

Overall Limit =Does Not Exist (DNE)

$\underset{x\to –1}{\lim }f\left(x\right)=\mathit{Does\; Not\; Exist}$