The Knowledge Web

The 0/0 Problem

When you try to solve a limit by Direct Substitution, you often hit a brick wall: $0/0$ . This doesn't mean the limit doesn't exist. It means there is a hidden hole in the algebra.

Algebraic X-Ray

1. The Problem

Direct substitution fails.

f(3) = 0/0

2. The X-Ray

Identify the problem term.

(x-3)(x+3) / (x-3)

3. The Solution

The hole is filled mathematically.

3 + 3 = 6

The Toolbox

1. Factoring

If you have a polynomial on top and bottom, factor them. Usually, the term causing the zero (like $x-3$ ) will appear in both and cancel out.

x^2 - a^2 = (x-a)(x+a)

2. Conjugates

If you see a square root (e.g., $\sqrt{x} - 2$ ), multiply the top and bottom by the conjugate ( $\sqrt{x} + 2$ ). This eliminates the root and reveals the hidden cancellation.