Consider the function $$f(x) = \frac{x^2-1}{|x-1|}\text.$$ This function is defined for all $x\neq 1$, so it is interesting to examine what happens around $x=1$. In the table below we fill in $x$ values near $1$:

We see that the function values to the left of the table approach $-2$, while on the right side of $1$ it seems that the limit is $2$.

In other words, for $x\in \left( a - \delta, a\right)$ the function value $f(x)$ is close to $L$. We write this in symbols as $$\lim_{x \uparrow a} f(x) = L\text.$$ Another conventional notation for the left limit is $\displaystyle \lim_{x\to a^-} f(x)\text.$ Similarly, we define the right limit of a function at a point.

We denote this as $$\lim_{x \downarrow a} f(x) = L\text.$$ Another conventional notation for right limit is $\displaystyle\lim_{x\to a^+} f(x)\text.$

In our notation, the example can be written as $$\lim_{x\uparrow 1} \frac{x^2-1}{|x-1|} = -2 \quad\text{ and }\quad \lim_{x\downarrow 1} \frac{x^2-1}{|x-1|} = 2$$