Approaching to a number and limit Consider $f(x)$ function . We want to calculate $\lim_{x \to 3}f(x)$. So for left limit , we approach to $3$ and then compute $f(2.9) , f(2.99) , f(2.999)$ and so on . Now there is a weird thing . It is obvious that $2.9999.... = 3$ and also when we are talking about limit , point isn't important . In this case we don't take care about $f(3)$ but when we approach to $3$ infinitely , we get $3$ as $2.9999.... = 3$ ! . I'm very confused about these two concepts .
 A: No, in order to find that some real $l$ is the limit
$$
\lim_{x\to3^-}f(x)
$$
you don't compute $f(2.9)$, $f(2.99)$ and so on. And neither you compute $f(2.(9))$ (periodic $9$), because no assumption is made that $f$ is defined at $3$, nor the possible value of $f$ at $3$ is relevant for the existence of the limit.
Saying that
$$
\lim_{x\to3^-}f(x)=l
$$
means

for every $\varepsilon>0$ there exists $\delta>0$ such that, for $3-\delta<x<3$, it holds $|f(x)-l|<\varepsilon$.

You can compute $f(2.9...9)$, if you wish; it may give you an idea of what $l$ could be, but in general it won't.
A: The idea of using a limit $\mathit{x}\rightarrow \mathit{n}$ is that you approach to $\mathit{n}$ as close as possible, but you actually never reach it. 
Just forget that you are "computing" $f$ at every point because it is a missunderstanding. Imagine that you are moving along the graph of the function $f$, then when taking a limit you are getting as close as possible to a specific point without ever touching it, as the function does not need to be $defined$ at that point, or the image might be different than the limit itself.
Imagine the following case:
$$f(x) = \left \lbrace {x^2, x \not= 0 \atop
1 , x = 0}\right. $$ 
If you take $lim_{x\rightarrow0}f(x) = 0$ for both right and left limits, but the actual image is $f(0)=1$.
When one has the equallity between right limit, left limit and image at a certain point in a function, we then say that the function is $continuous$, but any function that is not continuous still has limits.
I hope I clarified that to you.
edit: keep in mind that when talking about real numbers, between any two numbers there is an infinity of more numbers, it doesn't matter how close you try to imagine them to be, and that is the idea exploited by the limit.
