Overview
This lecture covers fundamental techniques for computing limits, emphasizing straightforward cases and strategies for more challenging problems like the 0/0 indeterminate form.
Basic Limit Properties
- The limit as X approaches a of a constant C is C.
- The limit as X approaches a of X is a.
- The limit of a constant times a function is the constant times the limit of the function.
- The limit of a sum or difference is the sum or difference of the individual limits, if those limits exist.
- The limit of a product is the product of the individual limits.
- The limit of a quotient is the quotient of the individual limits, provided the denominator's limit is not zero.
- The limit of a function raised to a positive integer power is the limit of the function raised to that power.
- The limit of an nth root is the nth root of the limit, if the result exists (especially if n is even, the limit must be non-negative).
Limits Involving Compositions and Trig Functions
- The limit of a composition f(g(x)) as x approaches a is f(lim g(x)) if limits exist.
- For trig functions, if a is in the domain, the limit as X approaches a is simply the function evaluated at a (e.g., limit as X→a of sin(X) is sin(a)).
Limits of Polynomials and Rational Functions
- The limit of a polynomial as X approaches a is simply the value at that point (P(a)).
- For rational functions, evaluate at the point unless the denominator is zero; otherwise, further analysis is needed.
Evaluating Limits by Direct Substitution
- If substitution yields a finite number, that is the limit.
- Examples: substituting into polynomial, rational, and trig functions if within domain and denominator isn't zero.
Indeterminate Forms and Factoring Technique
- If substitution gives 0/0, factor numerator and denominator and cancel common terms, then re-evaluate the limit.
- Functions that are equal everywhere except possibly at one point (removable discontinuity) have the same limit at that point.
Example Problems
- When plugging in yields 0/0, factor both numerator and denominator, cancel, and substitute again.
- For rational functions, after cancelling, the simplified expression can be used to compute the limit.
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a specific point.
- Linear Operator — An operation that satisfies additivity and homogeneity (e.g., limit operator).
- Rational Function — A function that is the ratio of two polynomials.
- Indeterminate Form (0/0) — A limit expression where direct substitution gives 0/0, requiring further analysis.
- Removable Discontinuity — A point at which two functions differ but are otherwise identical on an interval.
Action Items / Next Steps
- Practice limit problems involving factoring to resolve 0/0 forms.
- Review rationalization techniques for simplifying expressions.
- Prepare for more advanced limit techniques in the next session.