Factorization vs Rationalization vs L'Hopital

Factorization is best for polynomial 0/0 forms. It's fast and algebraically clean. Example: $\frac{x^3-8}{(x^2-4)}$ at x=2 factors immediately.

Rationalization is best for expressions involving surds (square roots). Multiply by the conjugate to eliminate the radical. Example: (sqrt(1+x)-1)/x — multiply by (sqrt(1+x)+1).

L'Hopital's Rule is the universal method but often the slowest. It works for any 0/0 or infinity/infinity form but may require multiple applications.

Standard Limits vs Taylor Expansion

Standard limits are one-step solutions for simple expressions. Use them when you can directly match the pattern $\frac{sin(kx}{(kx)}$, (e^(f(x))-1)/f(x), etc.).

Taylor expansion is more powerful for complex combinations. It handles expressions like $\frac{sin x - x + x^3/6}{x}$^5 in one step where L'Hopital would need five applications.

When to Use Which:

• Polynomial ratio: Factorization first
• Surd present: Rationalization
• Single trig/exponential function: Standard limits
• Multiple functions combined: Taylor expansion
• Nothing else works: L'Hopital's Rule
• Oscillatory function times vanishing: Sandwich Theorem
• Base -> 1, exponent -> infinity: 1^infinity formula
• Root existence: Intermediate Value Theorem