Question:

Evaluate the following limit algebraically:

Answer:

To find the limit, we can't simply substitute $x = 2$ into the expression, because it would result in an indeterminate form of $\frac{0}{0}$. Instead, we need to manipulate the expression algebraically to simplify it before taking the limit.

Let's factor the numerator and denominator of the expression:

Now, we can cancel out the common factor of $(x - 2)$ in the numerator and denominator:

Since the denominator still evaluates to zero at $x = 2$, we need to simplify further. Notice that the numerator of the expression can be factored as well:

Now, we can cancel out the factor of $(x - 2)$:

Finally, we can substitute $x = 2$ into the simplified expression:

Therefore, the limit of the given expression, as $x$ approaches 2, is equal to $-6$.