Overview
This lecture covers advanced techniques for computing limits, including factoring, rationalizing denominators or numerators, and applying the squeeze theorem to challenging limit problems.
Factoring to Resolve 0/0 Limits
- Simple limits can often be found by direct substitution if no division by zero occurs.
- If substitution leads to 0/0, try factoring the numerator and denominator to cancel common terms.
- For higher-degree polynomials (like cubics), use long division to factor when a root is known.
- Cancel the problematic factor, and then substitute the value to find the limit.
Rationalizing Denominators and Numerators
- Rationalizing helps when the denominator or numerator contains irrational terms (like square roots).
- Multiply by the conjugate (change the sign between terms) to eliminate square roots from the denominator.
- This technique also works for limits involving square roots and subtraction that lead to 0/0 forms.
- After rationalizing, simplify and cancel terms causing 0/0, then substitute to find the limit.
Squeeze Theorem for Limits
- The squeeze theorem applies when a function is trapped between two others that share the same limit.
- If ( h(x) \leq f(x) \leq g(x) ) and both ( h(x) ) and ( g(x) ) approach the same limit ( L ), then ( f(x) ) also approaches ( L ).
- Useful for limits involving oscillating functions (e.g., ( x \sin(1/x) )) or trigonometric functions.
- Bound the function using inequalities and show both bounds approach the same value as ( x ) approaches the limit point.
Key Terms & Definitions
- 0/0 Form — An indeterminate limit where substitution yields zero in both numerator and denominator.
- Factoring — Breaking down expressions into products of simpler expressions to simplify limits.
- Rationalizing — Multiplying by a conjugate to clear square roots from denominators or numerators.
- Conjugate — A binomial with the same terms and an opposite sign (e.g., ( a-b ) and ( a+b )).
- Squeeze Theorem — A method for determining limits when a function is bounded by two functions converging to the same limit.
Action Items / Next Steps
- Practice factoring and long division for polynomials in limit problems.
- Complete exercises on rationalizing denominators/numerators for limits involving square roots.
- Work through problems using the squeeze theorem, especially with trigonometric or oscillating functions.