Question:

Find the limit algebraically:

Answer:

To find the limit algebraically, we can attempt direct substitution by plugging in $x=2$ into the expression:

However, direct substitution results in an indeterminate form of $\frac{0}{0}$. This suggests that we need to manipulate the expression in order to simplify it and find the limit.

Factoring the numerator and denominator, we have:

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

We still have an indeterminate form of $\frac{0}{0}$, so we need to further simplify the expression. We can do this by factoring the quadratic in the numerator:

Now, we can cancel out the common factors of $(x+2)$:

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

Therefore, the limit of $\frac{{x^3 - 8}}{{x^2 - 4x}}$ as $x$ approaches $2$ is $-2$.