# What is the limit of (x^2-4)/(x-2) as x approaches 2?

Oct 18, 2014

A direct substitution results in the indeterminate form $\frac{0}{0}$. We should then try to simplify the function.

In this example we see that the numerator is a difference of perfect squares.

Remember factoring rules back to Algebra I.

$\left({a}^{2} - {b}^{2}\right) = \left(a - b\right) \left(a + b\right)$

In this example

$\left({x}^{2} - 4\right) = \left({x}^{2} - {2}^{2}\right) = \left(x - 2\right) \left(x + 2\right)$

${\lim}_{x \to 2} \frac{{x}^{2} - 4}{x - 2} = {\lim}_{x \to 2} \frac{\left(x - 2\right) \left(x + 2\right)}{x - 2}$

Cancel out the factor $\left(x - 2\right)$

${\lim}_{x \to 2} \left(x + 2\right)$

Direct substitution

$\left(2 + 2\right) = 4$