Overview
This lecture covers the fundamental properties of limits, including rules for evaluating the limits of constants, sums, differences, products, quotients, powers, and roots of functions.
Limits of Constants and Variables
- The limit as x approaches c of a constant k is k.
- The limit as x approaches c of x is c.
Sum and Difference Properties
- The limit of [f(x) + g(x)] as x approaches c is the sum of their limits: L + M.
- The limit of [f(x) - g(x)] as x approaches c is the difference of their limits: L - M.
Scalar Multiple Property
- The limit of k * f(x) as x approaches c is k times the limit of f(x): kL.*
Product and Quotient Properties
- The limit of [f(x) * g(x)] as x approaches c is the product of their limits: L * M.
- The limit of [f(x) / g(x)] as x approaches c is the quotient of their limits: L / M, provided M ≠ 0.
Power and Root Properties
- The limit of [f(x)]ⁿ as x approaches c is Lⁿ, where n is a positive integer.
- The limit of the nth root of f(x) as x approaches c is the nth root of L.
- If n is even, L must be positive; if n is odd, L can be any real number.
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a certain point.
- Constant Function — A function where every output is the same value.
- Scalar Multiple — A constant k multiplied by a function.
- Product Rule (Limits) — The limit of a product is the product of the limits.
- Quotient Rule (Limits) — The limit of a quotient is the quotient of the limits, if the denominator's limit is not zero.
- Power Rule (Limits) — The limit of a function to the nth power is the nth power of the function’s limit.
- Root Rule (Limits) — The limit of a root is the root of the limit, with restrictions depending on n and L.
Action Items / Next Steps
- Practice applying each limit property to example functions.
- Review cases where the denominator's limit is zero or the nth root conditions are not met.