Approximate the limit with a numerical approach. Choose a value near the limit and substitute it into the expression. Evaluate using a calculator. For example: the limit as x approaches two for a given expression may be approximated by substituting x with the value of 1.9, then 1.99, then 1.999. You may then approach the limit from the other side, using 2.1, 2.01 and 2.001. Try to determine if the limit of the expression converges to an obvious value. This step is optional, but may prove helpful.
Determine if the expression may be simplified. Typical limit problems take the form of a rational expression, so simplify by expansion and collection of like terms in both the numerator and denominator of your expression. If there is a sum or difference of like terms, combine them into one term if possible.
Cancel any common factors in the numerator and denominator of your rational expression. First, fully factor the numerator and denominator of your rational expression. This may require determining the greatest common factor, factoring a trinomial into binomials or factoring a polynomial of higher degree. Cancel any factors occurring in both the numerator and denominator of the rational expression.
Substitute the unknown variable of your now-simplified expression with the limit value. Evaluate the expression with this value. The result will be either a finite value, or will approach infinity. If you performed step one, your numerical calculations should have approached the same finite value, or become arbitrarily large, in these two cases respectively. If your numerical approach is consistent with your final answer, your work was probably correct.