Mathematics

Problem

Let

f(x)=\frac{x^3-8}{x^2-4}

. Both the numerator and denominator equal zero when

x=2

. a. Part 1: Calculate

\displaystyle \lim_{x\to 2} f(x)

. Present all algebraic work required to reach your final answer.

Intuitive explanation

This is a removable discontinuity problem—a chance to see what a function "wants" to be even when it looks undefined at one point. Since both numerator and denominator vanish at $x=2$ , you can factor both and cancel the common factor before evaluating. The key insight is that factoring reveals the hidden linear factor that's causing the trouble, letting you simplify to a form where $x=2$ is no longer problematic.

Intuitive explanation

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