Evaluate $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{x^2-4}{x-2}}$

The limit of $x$ square minus four divided by $x$ minus $2$ should be calculated as the value of $x$ approaches $2$ in this calculus problem. Firstly, let us try to find the limit by the direct substitution method.

So, substitute $x$ is equal to $2$ directly in the rational function to find the limit of the given function.

$=\,\,$ $\dfrac{2^2-4}{2-2}$

$=\,\,$ $\dfrac{2 \times 2-4}{2-2}$

$=\,\,$ $\dfrac{4-4}{2-2}$

$=\,\,$ $\dfrac{0}{0}$

It is evaluated that the limit of $x$ squared minus $4$ divided by $x$ minus $2$ is undefined as the value of $x$ is closer to $2$. The indeterminate form indicates that evaluating the limit by the direct substitution method is not suitable.

Methods

Factorization

Learn how to find the limit of $x$ squared minus $4$ divided by $x$ minus $2$ by factorisation as the value of $x$ tends to $2$.

L’Hôpital’s rule

Learn how to calculate the limit of $x$ square minus $4$ divided by $x$ minus $2$ by L’Hospital’s rule as the value of $x$ is closer to $2$.