What is the significance of limit and why is there always a difference between the value of x and the limiting value? Let $x$ be an independent variable and $y = f(x) = (x^2-1)/(x-1)$. If we draw the graph,then at $x = 1$, we cannot draw any point as y will be equal to $\frac00$,an indeterminate form. My book says if, without directly taking $x = 1$, $x$ is taken close to $1$, $f(x)$ also approaches a certain value, let $a$. Thus the value of $f(x)$ becomes close to $a$, when $x$ approaches $1$; limit of $f(x)$ as $x$ tends to $1 = a$.    
A simple bookish explanation. My problem is that what is the necessity of this concept. We know that when $x = 1$, $f(x) = \frac00$. What is the problem with $\frac00$? If there is any problem, how can the limit concept solve this? $x$ will always be $\frac00$, no matter what $f(x)$ is when $x$ approaches $1$. $a$, the limiting value, can never be equal to $\frac00$. I am unable to make any sense with the limit. Please help.   
My last question is my book says you cannot make the difference ie. magnitude of $x-a = 0$ hence you cannot make $x = a$, where $a$ is the limiting value. Why it is so? Help.

For 0/0, you have the problem that you can get anything, depending on how you approach it. For example, $$ \lim_{x \to 0} \frac{2x}x=2 $$ while $$  \lim_{x \to 0} \frac{-x}x=-1 $$ If we are close to x=0, the x's cancel and we get the value. However, if we set x=0 then 2*0=0 and -1*0=0 for the numerators, leaving us stumped.
In real analysis, you limits are defined much better (http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit). The limit exists if, given any distance to the point that the function is approaching, you can find a value of x close enough to force the function to be that close. 
Take the limit you gave as an example. If you say "Hey, I bet you can't get that limit within 1 unit of two," I can tell you to check when x=1.5 and then f[x]=2.5. If you ask for it to be within a thousandth, I can tell you to check x=1.0001. Lo and behold, it's within a thousandth.
