AP Calculus AB · Question of the Day

AP Calculus AB Question of the Day

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Friday, August 7, 2026

A model for efficiency uses $\lim_{x\to 2}\frac{x^2-4}{x^2-2x}$ . What is the limit?

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Question of the Day

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A model for efficiency uses $\lim_{x\to 2}\frac{x^2-4}{x^2-2x}$ . What is the limit?

$0$
$2$ (correct answer)
$4$
$\frac{1}{2}$
$\text{DNE}$

Explanation: Direct substitution gives 0/0, so we need to factor both numerator and denominator. The numerator x² - 4 is a difference of squares: (x - 2)(x + 2). The denominator x² - 2x factors as x(x - 2). Our limit becomes lim[x→2] [(x - 2)(x + 2)]/[x(x - 2)], and after canceling (x - 2), we get lim[x→2] (x + 2)/x. Substituting x = 2 gives (2 + 2)/2 = 4/2 = 2. Students sometimes factor only the numerator and forget about the denominator, or make errors when factoring out common terms. Always factor completely before canceling to avoid missing simplifications.