The function g(x) = (x² − 4)/(x − 2) is undefined at x = 2. What is the limit of g(x) as x approaches 2?
["Understanding the Function g(x) = (x² − 4)/(x − 2): Why It’s Undefined and What Its Limit Actually Is", "When analyzing rational functions, one common question arises: when does a function fail to be defined, and how do we interpret its behavior near those points? Take the function\n[ g(x) = \frac{x^2 - 4}{x - 2} ]\nThis function is undefined at ( x = 2 ), but does this make it useless at that point? Not at all. In this article, we explain why ( g(x) ) fails to be defined there and, more importantly, compute the limit as ( x ) approaches 2 — a concept vital in calculus and analysis.", "### Why Is g(x) Undefined at x = 2?", "A rational function like ( g(x) = \frac{x^2 - 4}{x - 2} ) is undefined wherever its denominator equals zero. Here, the denominator is ( x - 2 ), which becomes zero when ( x = 2 ). Since division by zero is undefined in mathematics, ( g(x) ) has a vertical asymptote or hole at ( x = 2 ). However, further inspection reveals more nuance — and a promising path to analyzing its behavior near that critical value.", "### What Is the Limit of g(x) as x Approaches 2?", "Even though ( g(2) ) does not exist, mathematicians often investigate:\n[ \lim_{x \ o 2} g(x) ]\nTo explore this, let’s simplify the expression algebraically.", "Note the numerator ( x^2 - 4 ) is a difference of squares:\n[ x^2 - 4 = (x - 2)(x + 2) ]\nSo we rewrite ( g(x) ) as:\n[ g(x) = \frac{(x - 2)(x + 2)}{x - 2} ]\nFor all ( x <br/>\neq 2 ), the ( x - 2 ) terms cancel:\n[ g(x) = x + 2 \quad \ ext{(when } x <br/>\ne 2\ ext{)} ]", "This simplified function ( g(x) = x + 2 ) is continuous and defined for all real numbers — including ( x = 2 ). In fact, directly substituting ( x = 2 ) gives:\n[ g(2) = 2 + 2 = 4 ]", "However, since the original expression is undefined at ( x = 2 ) (due to division by zero in the unsimplified form), the function itself is undefined there, but the limit exists and is equal to 4.", "### Interpreting the Limit", "While ( g(2) ) remains undefined,\n[ \lim_{x \ o 2} \frac{x^2 - 4}{x - 2} = \lim_{x \ o 2} (x + 2) = 4 ]\nThis means as values of ( x ) get arbitrarily close to 2 (but not equal to 2), the output of ( g(x) ) approaches 4. Graphically, the function has a removable discontinuity at ( x = 2 ), not a vertical asymptote — because the factor cancels, leaving a continuous curve with a “hole.”", "### Key Takeaways", "- ( g(x) = \frac{x^2 - 4}{x - 2} ) is undefined at ( x = 2 ) due to division by zero.\n- The simplified form ( g(x) = x + 2 ) (valid for ( x <br/>\ne 2 )) shows the function behaves like a line everywhere except at that point.\n- The limit as ( x \ o 2 ) exists and equals 4, even though ( g(2) ) is undefined.\n- This demonstrates how limits help analyze behavior near discontinuities — a cornerstone of calculus.", "### Final Note", "Understanding limits allows us to work with functions beyond points of discontinuity, transforming undefined expressions into meaningful values. In real applications, limits help model real-world phenomena where seemingly undefined behavior may vanish under careful analysis.", "In summary: The function ( g(x) = \frac{x^2 - 4}{x - 2} ) is undefined at ( x = 2 ) due to a zero denominator, but ( \lim_{x \ o 2} g(x) = 4 ), revealing a removable discontinuity and a well-defined limit that defines behavior near that critical point.", "---", "Keywords: function g(x) = (x² − 4)/(x − 2), undefined at x = 2, limit of g(x) as x approaches 2, purpose of limits, removable discontinuity, calculus concepts, rational functions.\nMeta Description: Discover why g(x) = (x² − 4)/(x − 2) is undefined at x = 2 yet has a limit of 4 as x approaches 2. Learn how limits clarify behavior near discontinuities in rational functions."]
