What is the point of finding a limit? Does limit give us a real/exact value? I wonder whether limit gives us an exact value. I mean look at this example:
$$\lim_{x\rightarrow 2} \frac{x^2 - 4}{x - 2}$$ 
is 4 , right?
But what if we do not use limit. Obviously at $x = 2$, it would be undefined.
So when x is approaching 2 we can say $f(x)$ is approaching 4. But when x is 2, we can not say it is 4, am I right? 
Then, what is the point in finding limit? Isn't it like you just "play" around a thing but not at the thing itself, no matter how close you are to the thing. Doesn't it feel like it is not real, not exact, or not concrete?
 A: Yes the limit is equal to $4$ since
$$\lim_{x \rightarrow 2}\frac{x^2-4}{x-2} = \lim_{x \rightarrow 2}\frac{(x+2)(x-2)}{x-2} =\lim_{x \rightarrow 2}x+2 = 4$$
This means that $f(x)$ can be made to be as close to 4 as desired by making $x$ sufficiently close to $2$. This is despite the fact that $f(2)$ is not defined.
A: Suppose that a car moves along a road such that, after $t$ s, it is at position $t^2$ m.
The average velocity between time $t=2$ s and time $t=x$ s is
$$\frac{x^2 - 4}{x - 2}$$
The limit shows that, if we measure the velocity at a small interval starting in $t=2$ s, the result will be close to 4 m/s. We say that 4 m/s is the velocity of the car at that instant.
The concept of instant velocity is one example of an application of limits.
A: It is perfectly exact; it's just not necessarily answering the question you want it to answer.
You're right that "$\lim_{x\rightarrow c}f(x)=L$" tells us nothing about $f(c)$, even whether $f(c)$ is defined. However, that's not the only use limits have! 
Example. Consider the parabola $f(x)=x^2$. At every point $(a, a^2)$ on the parabola, there is exactly one (non-vertical) line passing through $(a, a^2)$ and no other point on the parabola; this is the tangent line.$^*$ Via limits, we can find an exact expression for the tangent line through $(a, a^2)$ - it's $y=2ax-a^2$.
This is a nice example because we can prove, after the fact, that this line does in fact meet the parabola in exactly one point, and that it's the only (non-vertical) line through $(a, a^2)$ which does so. Moreover, the limit approach lets us calculate tangent lines to a huge variety of curves.
This is by far not the only example, but I think it's the most elementary one.

$^*$This isn't really the right way to define "tangent line", even for very nice functions; but it works here.
A: For your first and second questions: yes (as already said in the other answers). Let me consider the other question:
What is the point of finding a limit?
Look at the picture below.

Intuitively speaking, to $f$ be continuous at $2$ what should be the value of $f(2)$? Well, since $f(x)$ gets closer and closer to $4$ as $x$ moves closer and closer to $2$, it should be $4$ which is $\displaystyle\lim_{x\to 2}f(x)$.
This happens in general: $\displaystyle \lim_{x\to c} f(x)$ (if it exists) gives the value that $f$ should attains at $c$ in order to $f$ be continuous at $c$ (which is a real, exact and concrete thing, right?)
Remark: If such limit doesn't exist, it's not possible to make $f$ continuous at $c$ by defining (or redefining) the value of $f$ at $c$.
