The sequence 1/2, 1/3, 1/4, 1/5 and so on, has a limit of zero. The numbers are getting closer and closer to zero, but the limit is never actually reached. The limit of ((X + h)^2 - X^2) / h as h goes to zero is not so obvious. Especially troubling is the fact that if h reaches 0, the fraction is undefined. The "official" definition of a limit is: "The limit of f(X) is L, as X approaches h, if the distance between f(X) and L is as small as you want by making X close enough to h."
If the denominator of a fraction is going to zero, try canceling out a common factor to find the limit. For example the limit of (X^2 - 1) / (X - 1) as X goes to one is 2, because (X^2 - 1) / (X - 1) = (X + 1) / 1 = (1) + 1 = 2. If the limit involves something going to infinity, divide through by the degree of the largest polynomial. For example, the limit as X goes to infinity of (2X^2 -X - 1) / (X^2 + 1) = 2, because (2X^2/X^2 - X/X^2 -1/X^2) / (2X^2/X^2 + 1/X^2) = (2 - 1/X -1/X^2) / (1 + 1/X^2) = (2 - 0 - 0)/(1 + 0) = 2 as X goes to infinity.
If k and h are constants, the limit of k as X goes to h is k. The limit of X as X goes to h is h. The limit of hX as X goes to k is hk, and the limit as X goes to h of X^k is h^k. There is also a rule that should be part of the definition of limit. When we say something like "X goes to k," we seldom make a distinction between X approaching k from below or from above. If the difference exists, the limit is undefined.
If the limit of f(X) = k1 and the limit of g(X) = k2 as X approaches h, then the limit as X approaches h of [f(X) + g(X)] = k1 + k2, and the limit as X approaches h of [f(X) - g(X)] = k1 - k2. Furthermore, the limit as X approaches h of c X f(X) = c X k1 where c is a constant. Also the limit as X approaches h of [f(X) X g(X)] = k1 X k2, the limit as X approaches h of [f(X) / g(X)] = k1 / k2.