Lesson 2: Limits – Rules & Tricky Cases

1. Basic Limit Rules

If limits exist, they follow normal algebra:

• \(\lim (f + g) = \lim f + \lim g\)
• \(\lim (f · g) = (\lim f) · (\lim g)\)
• \(\lim (f/g) = (\lim f)/(\lim g)\) if \(\lim g ≠ 0\)
• \(\lim (c · f) = c · \lim f\) (where \(c\) is a constant)

Most continuous functions can be evaluated by direct substitution.

2. Indeterminate Forms & Simplification

\(0/0\) or \(∞/∞\) forms → simplify first.

Example: \(\lim_{x→2} (x² - 4)/(x - 2)\)

Direct: \(0/0\) → factor: \((x-2)(x+2)/(x-2) = x+2 (x≠2)\)

Limit = 4

3. One-Sided Limits

\(\lim_{x→a⁻} = \) from left, \(\lim_{x→a⁺} = \) from right

Limit exists only if both sides agree.

Example: \(f(x) = |x|/x\)

Left limit = -1, right limit = 1 → limit does not exist at \(x=0\)