Left Hand and Right Hand Limit

If $$f$$ is a real valued function, then $$x$$ can approach $$a$$ from two sides: the left side of $$a$$ and the right side of $$a$$. This is illustrated with the help of the diagram below.

Left Hand Limit
If $$x$$ approaches $$a$$ from the left side, i.e. from the values less than $$a$$, the function is said to have a left hand limit. If $$p$$ is the left hand limit of $$f$$ as $$x$$ approaches $$a$$, we write it as
\[\mathop {\lim }\limits_{x \to {a^ – }} f\left( x \right) = p\]

Right Hand Limit
If $$x$$ approaches $$a$$ from the right side, i.e. from the values greater than $$a$$, the function is said to have a right hand limit. If $$q$$ is the right hand limit of $$f$$ as $$x$$ approaches $$a$$, we write it as
\[\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = q\]

For the existence of the limit of a real valued function at a certain point, it is essential that both its left hand and right hand limits exist and have the same value.

In other words, if the left and right hand limits exist and
\[\mathop {\lim }\limits_{x \to {a^ – }} f\left( x \right) = \mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right)\],

then $$f$$ is said to have a limit at $$x = a$$.

On the other hand if both the left and right hand limits exist but
\[\mathop {\lim }\limits_{x \to {a^ – }} f\left( x \right) \ne \mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right)\],

then the limit of $$f$$ does not exist at $$x = a$$.