Overview
This lecture explains how to evaluate limits using properties of limits, applying them step-by-step to various algebraic combinations of functions.
Evaluating Limits with Constants and Sums
- The limit of a constant times a function equals the constant times the limit of the function: (\lim_{x \to a} [k f(x)] = k \lim_{x \to a} f(x)).
- Example: If (\lim_{x \to a} f(x) = 4), then (\lim_{x \to a} 4f(x) = 4 \times 4 = 16).
- The limit of a sum equals the sum of the limits: (\lim_{x \to a} [af(x) + bg(x)] = a \lim_{x \to a} f(x) + b \lim_{x \to a} g(x)).
- Example: If (\lim_{x \to a} f(x) = 4) and (\lim_{x \to a} g(x) = -3), then (\lim_{x \to a} [3f(x) + 5g(x)] = 3 \times 4 + 5 \times (-3) = -3).
Evaluating Limits with Products and Powers
- The limit of a product equals the product of the limits: (\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \times \lim_{x \to a} g(x)).
- Example: (4 \times -3 = -12).
- The limit of a function raised to a power is the limit raised to that power: (\lim_{x \to a} [g(x)]^4 = [\lim_{x \to a} g(x)]^4).
- The limit of the square root of a function is the square root of the limit: (\lim_{x \to a} \sqrt{f(x)} = \sqrt{\lim_{x \to a} f(x)}).
- Example: ((-3)^4 = 81), (\sqrt{4} = 2), so (81 \times 2 = 162).
Evaluating Limits with Quotients
- The limit of a quotient equals the quotient of the limits: (\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}) (if denominator ≠ 0).
- Alternatively, separate as (\lim_{x \to a} f(x) \times \lim_{x \to a} \frac{1}{g(x)}).
- Example: (4 \times \frac{1}{-3} = -\frac{4}{3}).
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a certain point.
- Constant Multiple Rule — The limit of a constant times a function is the constant times the function’s limit.
- Sum Rule — The limit of a sum is the sum of the limits.
- Product Rule — The limit of a product is the product of the limits.
- Quotient Rule — The limit of a quotient is the quotient of the limits, if the denominator's limit is not zero.
- Power Rule — The limit of a power is the power of the limit.
- Root Rule — The limit of a root is the root of the limit.
Action Items / Next Steps
- Practice evaluating limits using the constant, sum, product, quotient, power, and root rules.
- Show each step explicitly when solving limit problems for clarity.