Evaluate the limit \( \lim_{x o 2} rac{x^2 - 4}{x - 2} \).
["# Evaluate the Limit: ( \lim_{x \ o 2} \dfrac{x^2 - 4}{x - 2} )", "When encountering limits in calculus, one common challenge is evaluating expressions that appear undefined at first glance. A classic example is:", "[\n\lim_{x \ o 2} \dfrac{x^2 - 4}{x - 2}\n]", "At ( x = 2 ), both the numerator ( x^2 - 4 ) and the denominator ( x - 2 ) become zero, leading to the indeterminate form ( \dfrac{0}{0} ). This situation calls for simplification using algebraic techniques to reveal the true behavior of the function near the point.", "## Step-by-Step Evaluation", "### Step 1: Factor the Numerator\nNotice that ( x^2 - 4 ) is a difference of squares, which factors as:", "[\nx^2 - 4 = (x - 2)(x + 2)\n]", "Substituting this into the limit expression gives:", "[\n\lim_{x \ o 2} \dfrac{(x - 2)(x + 2)}{x - 2}\n]", "### Step 2: Simplify the Expression\nFor all ( x <br/>\ne 2 ), the term ( x - 2 ) in the numerator and denominator cancels out:", "[\n\lim_{x \ o 2} (x + 2)\n]", "### Step 3: Evaluate the Simplified Limit\nNow, substitute ( x = 2 ) into the simplified expression:", "[\n2 + 2 = 4\n]", "Thus, the limit exists and equals 4.", "[\n\lim_{x \ o 2} \dfrac{x^2 - 4}{x - 2} = 4\n]", "### Why This Approach Works\nBy factoring and simplifying, we resolve the indeterminate form and determine the limit by direct substitution. This method is efficient and avoids reliance on L’Hôpital’s Rule in simple polynomial rational limits.", "## Practical Implications and Further Learning\nUnderstanding how to evaluate such limits is foundational in calculus, especially when dealing with continuity and function behavior near points of discontinuity. This technique also extends to more complex rational functions and rational limits encountered in advanced mathematics.", "### Related Topics\n- Factoring polynomials\n- Indeterminate forms and limit laws\n- Continuity and removable discontinuities\n- Derivatives via limit definitions", "Mastering these concepts enhances problem-solving skills in analysis and prepares learners for real-world applications in engineering, physics, and economics.", "---", "By clearly analyzing indeterminate forms through algebraic simplification, we conclude that:", "[\n\lim_{x \ o 2} \dfrac{x^2 - 4}{x - 2} = \boxed{4}\n]"]
