# 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 $$x = a$$.

BENSI

July 30@ 8:28 amDear Sir,

Please check the RIGH HAND LIMIT. I think there is a clerical error as it is defined as “If x approaches a from the left side,” hope to be changed as from the right side.

eMathZone

July 30@ 2:45 pmThank you for that, we have now corrected it.