Limits & Continuity: every key term you need

Limit Definition

lim(x→a) f(x) = L means f(x) approaches L as x approaches a. Doesn't require f(a)=L; describes behavior near a.

One-Sided Limits

lim(x→a⁻) f(x): from left. lim(x→a⁺) f(x): from right. Limit exists if both equal.

Limit Laws

Sum: lim(f+g) = lim f + lim g. Product: lim(f·g) = lim f · lim g. Quotient: if lim g ≠ 0.

Indeterminate Forms

0/0, ∞/∞, 0·∞, ∞-∞. Require algebraic manipulation or L'Hôpital's rule to evaluate.

Continuity Definition

f continuous at a if: (1) f(a) defined, (2) lim(x→a) f(x) exists, (3) lim(x→a) f(x) = f(a).

Types of Discontinuity

Removable: hole (can redefine). Jump: left/right limits differ. Infinite: vertical asymptote.

Intermediate Value Theorem

If f continuous on [a,b] and N between f(a) and f(b), then ∃c in (a,b) with f(c)=N. Guarantees zeroes exist.

Infinite Limits

lim(x→a) f(x) = ∞ means f unbounded above as x→a. Vertical asymptote at x=a.

Limits at Infinity

lim(x→∞) f(x) describes end behavior. If lim = L, horizontal asymptote y=L. Polynomial: look at highest degree term.

L'Hôpital's Rule

For indeterminate 0/0 or ∞/∞: lim f/g = lim f'/g'. Differentiate numerator and denominator separately.

Unit 1 Summary

Limits describe function behavior near a point. Continuity requires limit equals function value. Asymptotes describe limits at infinity.

More AP Calculus BC guides

• Applications of Derivatives
• Parametric & Polar
• Infinite Series
• Differential Equations