Limits: Analytical Techniques & L'Hôpital's Rule
Limits are the foundational concept of calculus. To evaluate lim(x→c) f(x), first try direct substitution. If that gives an indeterminate form (0/0 or ∞/∞), factor and cancel, rationalize, or apply L'Hôpital's Rule. L'Hôpital's Rule states: if lim(x→c) f(x)/g(x) produces 0/0 or ∞/∞, then lim(x→c) f(x)/g(x) = lim(x→c) f'(x)/g'(x)—take the derivative of the numerator and denominator separately (NOT the quotient rule). Example: lim(x→0) (sin x)/x produces 0/0. Applying L'Hôpital's: lim(x→0) cos(x)/1 = cos(0) = 1. You may need to apply L'Hôpital's multiple times if the result is still indeterminate. Also essential: one-sided limits (approaching from the left, x→c⁻, versus the right, x→c⁺). A limit exists at a point only if both one-sided limits exist and are equal. This concept underlies continuity: a function is continuous at x = c if f(c) is defined, the limit exists, and lim(x→c) f(x) = f(c).