The function is undefined at \( oxed{x = 1} \), with a limit of \(-2\) as \( x o 1 \).

["# Understanding "The Function Is Undefined at ( x = 1 ), with a Limit of (-2) as ( x \ o 1 )"", "In calculus and real analysis, evaluating function behavior near a point is essential for understanding continuity, derivatives, and integrals. One common scenario involves functions where the value at a specific point is undefined, but a limit exists as we approach that point. A classic example is when ( \lim_{x \ o 1} f(x) = -2 ), yet ( f(1) ) is undefined—often denoted as ( x = 1 ). This article explores what this means, how limits relate to undefined points, and why such behavior is mathematically significant.", "## What Does It Mean for a Function to Be Undefined at ( x = 1 )?", "When we say a function ( f(x) ) is undefined at ( x = 1 ), we mean that ( f(1) ) does not have a valid numerical or real value. This can occur due to division by zero, a logarithm of a non-positive number, or other domain restrictions. For instance:", "[\nf(x) = \frac{x - 2}{x - 1}\n]", "At ( x = 1 ), the denominator becomes zero, making ( f(1) ) undefined (an indeterminate form). Despite this, we can investigate what values ( f(x) ) approaches as ( x ) gets close to 1.", "## The Limit Exists: What Does It Mean as ( x \ o 1 )?", "The limit ( \lim_{x \ o 1} f(x) = -2 ) tells us that as ( x ) approaches 1 from either the left or the right, the values of ( f(x) ) get arbitrarily close to (-2). Importantly:", "- The limit describes the behavior near ( x = 1 ), not at ( x = 1 ).\n- Even though ( f(1) ) is undefined (there may be a hole, jump, asymptote, or removable discontinuity), the limit reveals the function’s trend at that critical point.", "For the function ( f(x) = \frac{x - 2}{x - 1} ), compute the one-sided limits:", "[\n\lim_{x \ o 1^-} f(x) = \lim_{x \ o 1^-} \frac{x - 2}{x - 1} = \frac{-1}{0^-} = +\infty \quad \ ext{(not } -2\ ext{)}\n]", "This diverges, so not all functions with undefined values at a point have finite limits—or limits at all. The correct example where the limit is (-2) requires a more carefully designed function.", "### A Reasonable Example with a Finite Limit", "Consider:", "[\nf(x) = \frac{2x - 4}{x - 1}\n]", "At ( x = 1 ), denominator is zero; numerator is ( 2(1) - 4 = -2 ). This still suggests vertical asymptote behavior unless canceled.", "Try simplifying:", "[\nf(x) = \frac{2(x - 2)}{x - 1}\n]", "Still undefined at ( x = 1 ). For a limit of (-2), consider a rational function designed so numerator and denominator both approach values that yield a finite limit.", "Let:", "[\nf(x) = \frac{-2x + 2}{x - 1} = \frac{-2(x - 1)}{x - 1}\n]", "Cancel safely (for ( x <br/>\ne 1 )):", "[\nf(x) = -2, \quad \ ext{for } x <br/>\ne 1\n]", "Now ( f(x) ) is undefined only at ( x = 1 ), yet the limit as ( x \ o 1 ) is:", "[\n\lim_{x \ o 1} f(x) = -2\n]", "This illustrates a removable discontinuity—a function undefined exactly at a point but bounded and approaching a finite value near it.", "## Why This Matters in Mathematics", "Understanding limits at points of undefined functions is critical for:", "- Detecting removable discontinuities: A hole in a graph exists if ( \lim_{x \ o a} f(x) ) exists but ( f(a) ) is missing or differs.\n- Extending functions: Replacing undefined values with limits allows defining functions more continuously on broader domains.\n- Analyzing behavior in applications: In physics, engineering, and economics, thresholds or critical values are often studied via limits rather than pointwise function values.", "## Summary", "- The phrase “undefined at ( x = 1 ) with a limit of (-2)” indicates a removable discontinuity.\n- The limit exists and equals (-2) as ( x \ o 1 ), meaning values near ( x = 1 ) behave like ( -2 ) without touching it.\n- Such functions can be redefined at the point to become continuous, though the original function still lacks a value there.\n- Recognizing this behavior deepens understanding of function limits, continuity, and real-world modeling.", "Recall: A function’s limit near an undefined point does not depend on the function’s value at that point—but it reveals essential local behavior that guides analysis and application.", "## Further Reading", "- Continuity and discontinuities in calculus\n- Rational functions and limits\n- One-sided limits and discontinuity types", "---", "Understanding the distinction between defined function values and limit behavior equips learners and practitioners to interpret functions accurately in both theoretical and practical contexts. Whether graphing, solving equations, or modeling phenomena, mastering these concepts ensures robust mathematical reasoning."]