# What are common mistakes students make with limits?

##### 1 Answer
Apr 13, 2016

One common mistake is thinking that initial form $\frac{0}{0}$ implies that the limit does not exist.

#### Explanation:

For example:

Evaluate ${\lim}_{x \rightarrow 2} \frac{{x}^{2} - 4}{{x}^{2} - 2 x}$

$\textcolor{red}{\text{Student Error}}$
${\lim}_{x \rightarrow 2} \frac{{x}^{2} - 4}{{x}^{2} - 2 x} = \frac{{\left(2\right)}^{2} - 4}{{\left(2\right)}^{2} - 2 \left(2\right)} = \frac{0}{0} = \textcolor{red}{D N E}$

$\textcolor{b l u e}{\text{Correct solution}}$

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