One of the most powerful tools in calculus is the ability to evaluate limits algebraically. By manipulating equations and applying various algebraic techniques, we can determine the behavior of a function as it approaches a particular value. Let's explore some of the common methods used in evaluating limits:

1. Factoring: Factoring is often the first step in simplifying a complex expression. By factoring out common terms, we can cancel out any common factors and simplify the expression.

Example: Consider the limit of the function (x^2 - 4)/(x - 2) as x approaches 2. By factoring the numerator into (x + 2)(x - 2), we can cancel out the common factor (x - 2) and simplify the expression to (x + 2). Thus, the limit as x approaches 2 is 4.

2. Rationalizing: Sometimes we encounter expressions with radicals in the denominator. Rationalizing the denominator involves multiplying the numerator and denominator by an appropriate conjugate to eliminate the radical.

3. Simplifying: Simplifying an expression involves canceling out common factors, simplifying fractions, and combining like terms to obtain a more manageable expression.

Remember, when evaluating limits algebraically, always simplify as much as possible before substituting the desired value.

Keep practicing these techniques, and soon you'll be a master at evaluating limits algebraically!