Example 1: Left Hand Limit

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 have a negative sign ($\mathit{–}$) in the exponent of our $a–\mathit{value}$. So, we are being asked to find the Left-Hand Limit.

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

Step 3 ( LHL ): Begin 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 .

Notice that you do not care about any jumps in the graph as you do this. It is all one graph, $f\left(x\right)$. So, jump from one piece to the next and continue to follow the graph until you run into the “wall”.

The $y–\mathit{value}$ where you run into the wall is the answer to the question.

In this example we run into the “wall” at $y=–1$.

So, the answer to our limit question is $y=–1$.

Final Result:

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