AP Precalculus: Limits – Formulas & Laws

1. Limits – Definition & Graphical Meaning

\( \displaystyle \lim_{x \to a} f(x) = L \) means as \( x \) approaches \( a \), \( f(x) \) approaches \( L \)
One-sided limits:
- \( \displaystyle \lim_{x \to a^-} f(x) \): from the left
- \( \displaystyle \lim_{x \to a^+} f(x) \): from the right
Limit exists if left and right limits are equal and finite
Jump, infinite, or removable discontinuities: break, asymptote, hole, etc.

Asymptote: \( \displaystyle \lim_{x \to a} f(x) = \pm \infty \)
End behavior: \( \displaystyle \lim_{x \to \infty} f(x) \), \( \displaystyle \lim_{x \to -\infty} f(x) \)

Addition: \( \displaystyle \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \)
Subtraction: \( \displaystyle \lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) \)
Multiplication: \( \displaystyle \lim_{x \to a} [f(x) g(x)] = (\lim_{x \to a} f(x))(\lim_{x \to a} g(x)) \)
Division: \( \displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \) (if denominator \(\ne 0\))
Power: \( \displaystyle \lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n \)
Root: \( \displaystyle \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} \) (if root exists)

For polynomial \( p(x) \): \( \displaystyle \lim_{x \to a} p(x) = p(a) \)
For rational \( \frac{p(x)}{q(x)} \), \( q(a) \ne 0 \): limit is \( \frac{p(a)}{q(a)} \)
If direct substitution gives \( \frac{0}{0} \), factor and simplify