In calculus, evaluating limits algebraically is an important skill that allows us to determine the behavior of functions as they approach specific values. There are several algebraic techniques that can be used to evaluate limits, including direct substitution, factoring, and rationalizing.

Direct substitution is the simplest method and involves evaluating the function at the given value. For example, to find the limit of f(x) = (x^2 - 4)/(x - 2) as x approaches 2, we substitute x = 2 into the function and simplify the expression. In this case, the limit is 4.

Factoring can be helpful when dealing with functions that contain polynomials. Consider the limit of f(x) = (x^3 - 8)/(x - 2) as x approaches 2. By factoring the numerator as (x - 2)(x^2 + 2x + 4), we can cancel out the common factor of (x - 2) and evaluate the limit as x approaches 2 to get a result of 12.

Rationalizing is another technique that can be used to evaluate limits. For instance, let's find the limit of f(x) = sqrt(x^2 + 3) - x as x approaches 0. By multiplying the expression by its conjugate, sqrt(x^2 + 3) + x, we can eliminate the square root and simplify the expression. The limit in this case is -3/2.