f(x) has a vertical asymptote at x = -3, so the limits from each side will shoot to either plus or minus infinity. If you plug in a few values close to x = -3 from each side, say x = -3.01 from the left and x = -2.99 from the right, you'll see that the limit shoots to +∞ from the left and to -∞ from the right. Since the limits are not the same, there is no limit for f(x).
Use a graphing calculator or program to graph f(x) = x/(x+3) to confirm the behavior.
First of all, your statement about how to find a limit suggests that you really do not understand what a limit is and I suggest that you go back and look at the definition of a limit and MEMORIZE IT WORD FOR WORD! If you do not understand what a limit really is, you will never thoroughly understand calculus!
The trick with the problem you have is that the rational function is discontinuous at x=-3 where there is a 0 of the denominator. Therefore, it does not approach a limit...it blows up!