Left Hand and Right Hand Limit

If f is a real valued function, then x can approaches a from two sides, the left side of a and the right side of a. This is illustrated with the help of diagram as shown.


l-r-limit

Left Hand Limit:
If x approaches a from left side, i.e. from the values less than a, the function is said to have 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 left side, i.e. from the values greater than a, the function is said to have 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 limit of a real values function at a certain point, it is essential that it’s both left hand and right hand limits exists and have same value.
In other words if the left and right hand limits exists 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 left and right hand limit exists 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.

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