Definition of the Limit of a Function

Moving one step beyond limits of sequences, we now study the limit of a function f(x). We examine the behavior of f(x) as x approaches a arbitrarily closely.

As x approaches a arbitrarily closely, if f(x) approaches a fixed value α arbitrarily closely, we write
　lim_{x→a} f(x) = α. “x → a” means “x approaches a” (x need not equal a itself).

Limits of the Four Basic Operations

When lim f(x) = α and lim g(x) = β:

• lim (f ± g) = α ± β, lim (kf) = kα
• lim (fg) = αβ, lim (f/g) = α/β (β≠0)

For a continuous function, lim_{x→a} f(x) = f(a).

The 0/0 Indeterminate Form

When both the numerator and denominator are 0, direct substitution is impossible. Factor and cancel.
Example: lim_{x→2} (x²−4)/(x−2) = lim (x+2) = 4

Practice Problems

1. lim_{x→3} (x²−9)/(x−3)
2. lim_{x→0} (√(x+4)−2)/x
3. lim_{x→−1} (x²+3x+2)/(x+1)