# What exactly is a limit in calculus?

$f \left(x\right) = \frac{{x}^{2} - 1}{x - 1}$
Since its denominator is zero when $x = 1$, $f \left(1\right)$ is undefined; however, its limit at $x = 1$ exists and indicates that the function value approaches $2$ there.
lim_{x to 1}{x^2-1}/{x-1} =lim_{x to 1}{(x+1)(x-1)}/{x-1} =lim_{x to 1}(x+1)=2