Why does 0/0 mean you can factor and cancel

When a limit yields 0/0, the expression is not undefined; it merely indicates that algebraic simplification can expose the true behavior. The zeroes in numerator and denominator usually come from a common factor that vanishes at the limit point.

Example: \(\lim_{x\to2}\frac{x^2-4}{x-2}\)

Simplify the fraction by factoring and canceling:

1. 1Factor numerator: $x^2-4=(x-2)(x+2)$.
2. 2Write fraction as $\frac{(x-2)(x+2)}{x-2}$.
3. 3Cancel the common $(x-2)$ factor (valid for $x\neq2$).
4. 4Resulting expression: $x+2$.
5. 5Take the limit: $\lim_{x\to2}(x+2)=4$.

The cancellation is justified because we are interested in values arbitrarily close to 2, not at 2 itself. After canceling, the limit is easy to evaluate.

Typical situations where factoring works:

• Polynomials with a common linear factor.
• Difference of squares or cubes.
• Expressions that can be rewritten using known identities.

Common indeterminate forms and standard techniques:

Always verify that the factor you cancel truly appears in both numerator and denominator; otherwise you may remove essential behavior and obtain an incorrect limit.