Limits

$\lim_{x \to a} f(x) = L$ means $f(x)$ stays arbitrarily close to $L$ whenever $x$ is close enough to $a$ (with $x \neq a$). The value of $f$ at $a$ itself is irrelevant — a limit describes approach, not arrival.

Limit laws: limits distribute over sums, products, and quotients (nonzero denominator). Direct substitution works for polynomials and other continuous functions. An indeterminate $0/0$ form signals algebra: factor, cancel, or rationalize. The cornerstone trig limit is $\lim_{x \to 0} \frac{\sin x}{x} = 1$.

Compute $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$. Substitution gives $0/0$, so factor: $\frac{(x-3)(x+3)}{x-3} = x + 3$ for $x \neq 3$. Now substitute: the limit is $3 + 3 = 6$.