Question:

Find the limit algebraically:

Answer:

To find the limit of the given expression, we can plug in the value that $x$ is approaching and see what we get. However, when we substitute $x = -4$ directly into the expression, we encounter an indeterminate form of $\frac{0}{0}$.

Therefore, instead of performing direct substitution, we can use algebraic manipulation in order to simplify the expression and find the limit.

Step 1: Factorize the numerator as the difference of squares:

Step 2: Rewrite the expression:

Step 3: Cancel out the common factor of $x + 4$:

Now, we can take the limit as $x$ approaches $-4$ of the simplified expression:

Since we have eliminated the indeterminate form, we can now plug in $x = -4$ directly into the expression:

Therefore, the limit of the given expression as $x$ approaches $-4$ is $-8$.