Find the limit as \( x \) approaches 2 of the function \( f(x) = rac{x^2 - 4}{x - 2} \).

["# Find the Limit as ( x ) Approaches 2 of ( f(x) = \frac{x^2 - 4}{x - 2} )", "Understanding limits is fundamental in calculus, especially when analyzing function behavior at specific points. One common question in introductory calculus is: What is the limit of ( f(x) = \frac{x^2 - 4}{x - 2} ) as ( x ) approaches 2? While substituting ( x = 2 ) directly leads to an undefined form ( \frac{0}{0} ), careful analysis reveals insight into the function’s behavior near this point. This article explains how to find the limit, explores the simplification process, and discusses the implications in calculus.", "## Understanding the Function", "The function in question is:", "[\nf(x) = \frac{x^2 - 4}{x - 2}\n]", "At a glance, substituting ( x = 2 ) gives:", "[\nf(2) = \frac{2^2 - 4}{2 - 2} = \frac{0}{0}\n]", "This is an indeterminate form, meaning the function does not have a defined value at ( x = 2 ). However, limits capture the behavior of the function as ( x ) approaches 2, not necessarily the value at 2.", "## Factor the Numerator", "To resolve the indeterminate form, factor the numerator ( x^2 - 4 ), recognizing it as a difference of squares:", "[\nx^2 - 4 = (x - 2)(x + 2)\n]", "Rewriting the function:", "[\nf(x) = \frac{(x - 2)(x + 2)}{x - 2}\n]", "For all ( x <br/>\neq 2 ), the ( x - 2 ) terms cancel:", "[\nf(x) = x + 2\n]", "This simplified expression is valid everywhere except ( x = 2 ), but it reveals a key insight: the function behaves exactly like ( x + 2 ) near ( x = 2 ), except at ( x = 2 ) itself.", "## Evaluate the Limit", "Now, compute the limit using the simplified expression:", "[\n\lim_{x \ o 2} f(x) = \lim_{x \ o 2} (x + 2) = 2 + 2 = 4\n]", "Even though ( f(2) ) is undefined, the limit exists and equals 4. This means as ( x ) gets arbitrarily close to 2 from either side, ( f(x) ) approaches 4.", "## Graphical Interpretation", "If we graph ( f(x) = \frac{x^2 - 4}{x - 2} ), at ( x = 2 ) there appears to be a “hole.” A complete graph includes the point ( (2, 4) ) by defining the limit, even if the function is not defined there due to cancellation and removable discontinuity.", "## Why the Limit Exists Despite Undefined Point", "The existence of the limit relies on continuity of the simplified function ( x + 2 ) around ( x = 2 ). Since the original function matches ( x + 2 ) everywhere except ( x = 2 ), and the limit confirms ( \lim_{x \ o 2} f(x) = 4 ), we conclude the limit exists and is well-defined.", "## Common Errors to Avoid", "- Substituting directly at ( x = 2 ), leading to ( \frac{0}{0} ), which fails due to indeterminate form.\n- Ignoring simplification when factoring reveals the function’s true behavior.\n- Assuming undefined = non-existent limit, when limits focus on approach, not point evaluation.", "## Conclusion and Key Takeaway", "The limit of ( f(x) = \frac{x^2 - 4}{x - 2} ) as ( x ) approaches 2 is:", "[\n\lim_{x \ o 2} \frac{x^2 - 4}{x - 2} = 4\n]", "This result demonstrates how factoring and simplification unlock limit evaluation even when direct substitution fails. Understanding such techniques is essential for mastering calculus and analyzing rational functions near discontinuities.", "---", "Keywords: limit, function limit, ( \lim_{x \ o 2} ), ( \lim_{x \ o 2} \frac{x^2 - 4}{x - 2} ), removable discontinuity, simplification, calculus basics, limit rules, discontinuities in rational functions.", "This SEO-optimized article clarifies the process of finding the limit, addresses common misunderstandings, and emphasizes both algebraic technique and conceptual understanding—critical for students and anyone learning calculus limits."]