1. Why Limit Laws Matter
The limit laws allow us to break complicated limits into smaller, simpler parts. Instead of evaluating a limit directly, we apply rules that handle sums, products, quotients, and powers smoothly.
These rules work whenever the individual limits exist.
2. Basic Limit Laws
These laws describe how limits behave under common operations.
2.1. 1. Sum Law
\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)
2.2. 2. Difference Law
\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)
2.3. 3. Constant Multiple Law
\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)
2.4. 4. Product Law
\lim_{x \to a} [f(x)g(x)] = (\lim_{x \to a} f(x)) (\lim_{x \to a} g(x))
2.5. 5. Quotient Law
\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)} \quad \text{provided the denominator limit is not zero}
2.6. 6. Power Law
\lim_{x \to a} [f(x)]^n = (\lim_{x \to a} f(x))^n
2.7. 7. Root Law
\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}
3. Important Special Limits
These limits occur often and act as building blocks for more complex expressions.
3.1. Limit of a Constant
\lim_{x \to a} c = c
3.2. Limit of x
\lim_{x \to a} x = a
3.3. Limit of a Polynomial
To find the limit of a polynomial, just substitute the value directly:
\lim_{x \to a} P(x) = P(a)
4. Using Limit Laws Together
Most real problems require applying multiple laws at once: breaking the expression into smaller parts, applying a law to each part, and recombining the results.
4.1. Example
Evaluate:
\lim_{x \to 3} (2x^2 - 5x + 4)
Use limit laws to substitute directly:
2(3^2) - 5(3) + 4 = 18 - 15 + 4 = 7
5. When Limit Laws Cannot Be Used Directly
Limit laws need the individual limits to exist. If a function is undefined or behaves wildly near a point, you may need algebraic simplification or techniques like rationalizing, factoring, or special rules (like L'Hospital's Rule).