Intro to Limits & Continuity
Limits are the foundation of calculus. They let us describe what a function approaches near a point — even if the function isn't defined there.
Key Concepts
What Is a Limit?
lim[x→a] f(x) = L means f(x) gets closer and closer to L as x approaches a. The limit describes behavior near a, not necessarily at a. The function value f(a) may differ from the limit.
One-Sided Limits
Left-hand limit: x approaches a from the left (x→a⁻). Right-hand limit: x approaches from the right (x→a⁺). The full limit exists only when both one-sided limits are equal.
Continuity
A function f is continuous at a if: 1) f(a) is defined, 2) lim f(x) exists as x→a, 3) lim f(x) = f(a). Polynomials, sin, cos, e^x are continuous everywhere.
Limit Laws
Limits distribute over + − × ÷. lim[x→a](f±g) = lim f ± lim g. lim(f·g) = (lim f)(lim g). For indeterminate forms 0/0, try factoring, rationalizing, or L'Hôpital's rule.
Live Python Practice
Interactive Lab
Check Your Understanding
lim[x→2] (x²−4)/(x−2) equals:
A function f is continuous at a if:
lim[x→0] sin(x)/x equals: