Why we take right hand and left hand limit

[otomanb]

Why we take right hand and left hand limit of a function?
what actually we are trying to find on that time? asymptotes or ...?

 

[Simon Bridge]

The limit may be different from different directions.
eg. Unit step function

$$u(x-x_0)=<br /> \begin{cases} <br /> 0 \; : \; \; x \leq x_0,\\<br /> 1 \; : \; \; x > x_0 \<br /> \end{cases}$$

(edit: TeX redone thanks spamiam xD )

thus:
$$\lim_{x \rightarrow x_0}{u(x-x_0)}$$
... is either 0 or 1 depending on which direction you approach it from.

(Though, in this case, the function is defined for x=x₀, sometimes the function is not defined at the limit but well-behaved on either side of it.)

So the general answer to your question is that we take left and right-hand limits to investigate the behavior of a function on either side of the limit.

 

[spamiam]

The nice thing about the real numbers is that the limit of a function at a point exists if and only if the left- and right-hand limits exist and are equal. Note that in higher dimensions, we have to consider infinitely many "directions" from which we can approach the point of interest, whereas for the reals there are only 2.

@Simon Bridge--Like this:

$$u(x-x_0)=<br /> \begin{cases} <br /> 0 \quad \text{if} \; \; x \leq x_0,\\<br /> 1 \quad \text{if} \; \; x > x_0 \<br /> \end{cases}$$

 

[Simon Bridge]

@spamiam: Thanx xD