Overview
This lecture covers the fundamental theorems of limits, standard techniques for evaluating limits, and handling indeterminate forms using factorization, rationalization, and related strategies.
Fundamental Theorems on Limits
- If lim(x→a) f(x) = L and lim(x→a) g(x) = M, then both L and M are finite numbers.
- Limit law for addition/subtraction: lim(x→a) [f(x) ± g(x)] = lim(x→a) f(x) ± lim(x→a) g(x).
- Limit law for multiplication: lim(x→a) [f(x) × g(x)] = lim(x→a) f(x) × lim(x→a) g(x).
- Limit law for division: lim(x→a) [f(x)/g(x)] = lim(x→a) f(x) / lim(x→a) g(x), given lim(x→a) g(x) ≠0.
Direct Substitution and Indeterminate Forms
- If substitution gives a defined value, use direct substitution to evaluate the limit.
- If substitution leads to 0/0 or ∞/∞, the form is indeterminate and requires further techniques.
- Common indeterminate forms are 0/0 and ∞/∞.
Techniques to Solve Indeterminate Forms
- Factorization: Factor expressions to cancel problematic terms and resolve 0/0 forms.
- Rationalization: Multiply by a conjugate to simplify limits involving square roots.
- Use the "a²-b² = (a-b)(a+b)" formula to handle differences of squares.
- For trigonometric limits, use inequalities such as -1 ≤ cos(θ) ≤ 1 for all θ.
Examples & Applications
- lim(x→0) x² sin(1/x) = 0 by applying limit laws and evaluating boundedness.
- Factorize both numerator and denominator for expressions like lim(x→1) (x²-1)/(x-1).
- Rationalize to solve, e.g., lim(x→1) [√(25-x²) - √24]/(x-1).
- When stuck with 0/0 form after all algebraic manipulations, L'Hospital's Rule may be applied.
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches some value.
- Indeterminate Form — An expression where direct substitution does not yield a definitive value (e.g., 0/0).
- Factorization — Rewriting an expression as a product of its factors.
- Rationalization — Multiplying numerator and denominator by a conjugate to remove roots.
- L'Hospital's Rule — Differentiation technique used to solve 0/0 or ∞/∞ indeterminate forms.
Action Items / Next Steps
- Practice evaluating limits using direct substitution, factorization, and rationalization.
- Review examples of solving limits resulting in indeterminate forms.
- Prepare for questions involving the application of fundamental limit theorems.