Usually the best thing to do when you are dealing with a new limit is to plug in the value of interest. Your value of interest is the value the limit is "going to." For example, mathematicians often write limits using the phase "the limit as [this function] goes to a." Here, "a" is your value of interest. Put this value into the function and see what happens. If you get a number, you have solved the limit. If the limit goes to infinity or negative infinity, you are finished as well because in this case, there is no solution.
Many limit problems require you to simplify them before you can find the solution. The type of simplification you should use can only be determined on a case-by-case basis. Look closely at the limit and see if you can find anything to take out or reduce. Some common techniques are factoring, finding a common denominator for fractions and multiplying by the conjugate.
Rewriting a limit using its true definition may help you find the solution. An example of this situation is when you are finding the limit of a piecewise function. In this situation, it is best to rewrite the limit of the piecewise function in terms of the definition of a piecewise function. This can help you look more easily at the limits of the individual pieces of the function.
L'Hopital's rule may allow you to rewrite a problematic limit -- such as a limit that gives you division by zero or infinity dividing infinity when you plug in the value of interest -- as a different limit. Applying L'Hopital's rule is as easy as taking the derivative of both the numerator and denominator of the limit. This will give you a new limit, but the answer to this new limit will be the same as the answer to the original limit. In other words, you can change the limit problem to another limit problem that may be simpler to solve.