Limits

A limit describes the value a function approaches as the input approaches a particular point. Fundamental in calculus, derivatives, and integrals.

Key Concepts:

1. Notation: $$\lim_{x \to a} f(x) = L$$x→alim​f(x)=L
2. Properties:
   • $\lim_{x \to a} [f(x) ± g(x)] = \lim f(x) ± \lim g(x)$limx→a​[f(x)±g(x)]=limf(x)±limg(x)
   • $\lim_{x \to a} [f(x)·g(x)] = \lim f(x) · \lim g(x)$limx→a​[f(x)⋅g(x)]=limf(x)⋅limg(x)
   • $\lim_{x \to a} [f(x)/g(x)] = (\lim f(x))/(\lim g(x))$limx→a​[f(x)/g(x)]=(limf(x))/(limg(x)) if $\lim g(x)≠0$limg(x)=0
3. Techniques:
   • Direct substitution
   • Factoring
   • Rationalization
4. Applications: Calculus, continuity, derivatives, integration

Example Problem:$$\lim_{x \to 2} (x^2 – 4)/(x-2) = \lim_{x \to 2} (x-2)(x+2)/(x-2) = \lim_{x \to 2} x+2 = 4$$x→2lim​(x2−4)/(x−2)=x→2lim​(x−2)(x+2)/(x−2)=x→2lim​x+2=4