Question

What is a two sided limit?

Answer 1

From the question we have been asked about the two sided limit. To solve this question we will take the help of limits and continuity and we will use its property and explain all its possible conditions and we will prove the left hand limit and right hand limit of a function and solve this question. So, our solution will be as follows.

Complete step by step solution:
We are talking here about the limit of a function $f(x)$ as its argument $x$ approaches a concrete real number A within its domain.
It's not a limit when an argument tends to infinity.
The argument $x$ can tend to a concrete real number A in several ways:
(a) $x \to A$ while $x< A$, denoted sometimes as $x \to {{A}^{-}}$
(b) $x \to A$ while $x> A$ denoted sometimes as $x \to {{A}^{+}}$
(c) $x \to A$ without any additional conditions
All the above cases are different and conditional limits (a) and (b), when
$\Rightarrow x \to {{A}^{-}}$ and $x \to {{A}^{+}}$, might or might not exist independently from each other and, if both exist, might or might not be equal to each other.
Of course, the unconditional limit (c) of a function when $x \to A$ exists. The other two, the conditional ones, exist as well and are equal to the unconditional one.
The limit of $f(x)$ when
$\Rightarrow x \to {{A}^{-}}$
is a one-sided (left-sided) limit.
The limit of $f(x)$ when
$\Rightarrow x \to {{A}^{+}}$ is also a one-sided (right-sided) limit.
If they both exist and equal, we can talk about two-sided limit
So, two-sided limit can be defined as follows:
If,
$\Rightarrow L=\displaystyle \lim_{x \to {{A}^{-}}}$ $f(x)$ exists and
$\Rightarrow R=\displaystyle \lim_{x \to {{A}^{+}}}$ $f(x)$ exists and
$\Rightarrow R=L$
Then, $R=L$ is called a two sided limit.

Note: Students should have good knowledge in the concept of limits and continuity. We should note that if $\Rightarrow R=L$ then only it is said to be a two sided limit. If $\Rightarrow R\ne L$ then it is not a two side limit instead it will be having a two one sided limit.