Evaluate the limit \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \).

["Evaluate the Limit: ( \lim_{x \ o 2} \frac{x^2 - 4}{x - 2} )", "When approaching calculus problems involving limits, one fundamental concept is evaluating the limit as a variable approaches a specific value. A commonly encountered example is:", "[\n\lim_{x \ o 2} \frac{x^2 - 4}{x - 2}\n]", "At first glance, direct substitution of ( x = 2 ) leads to an indeterminate form:", "[\n\frac{2^2 - 4}{2 - 2} = \frac{0}{0}\n]", "Indeterminate forms indicate that the limit cannot be determined by simple substitution. Instead, mathematicians use algebraic simplification and factorization to resolve such expressions.", "### Step 1: Factor the Numerator", "Notice that the numerator ( x^2 - 4 ) is a difference of squares, which can be factored:", "[\nx^2 - 4 = (x - 2)(x + 2)\n]", "Substituting this factorization into the original expression, we rewrite the limit as:", "[\n\lim_{x \ o 2} \frac{(x - 2)(x + 2)}{x - 2}\n]", "### Step 2: Simplify the Expression", "For all ( x <br/>\neq 2 ), the ( x - 2 ) terms in the numerator and denominator cancel out:", "[\n\lim_{x \ o 2} (x + 2)\n]", "### Step 3: Evaluate the Simplified Limit", "Now, evaluate the limit of the simplified expression as ( x ) approaches 2:", "[\n\lim_{x \ o 2} (x + 2) = 2 + 2 = 4\n]", "### Conclusion", "Thus, the original limit evaluates to:", "[\n\lim_{x \ o 2} \frac{x^2 - 4}{x - 2} = 4\n]", "This result demonstrates a key principle in calculus: understanding indeterminate forms and applying algebraic techniques to simplify limits effectively. Recognizing the difference of squares and canceling common factors are essential tools in evaluating limits smoothly and accurately.", "Using this method, not only do we solve the current problem, but we also gain a deeper appreciation for limit evaluation techniques widely applicable in mathematical analysis.", "---", "Keywords: limit evaluation, indeterminate form, ( \lim_{x \ o 2} \frac{x^2 - 4}{x - 2} ), factorization, difference of squares, calculus tutorial, limit simplification.", "Meta Description: Learn how to evaluate ( \lim_{x \ o 2} \frac{x^2 - 4}{x - 2} ) using algebraic factorization and simplification—step-by-step guide with calculus insight."]