Limits & Continuity: every key term you need (+ practice quiz)

More AP Calculus BC guides

• Derivatives
• Applications of Derivatives
• Integration
• Applications of Integrals
• Parametric & Polar

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.

Epsilon-Delta Statement

lim(x→a) f(x)=L means for every ε>0 there is δ>0 with |f(x)-L|<ε whenever 0<|x-a|<δ. The δ is chosen after ε, never before.

Squeeze Theorem Setup

If g(x) ≤ f(x) ≤ h(x) near a and lim g = lim h = L, then lim f = L. Typically used when f contains a bounded oscillating factor like sin(1/x).

Oscillating Discontinuity

f(x)=sin(1/x) has no limit at 0 because it takes every value in [-1,1] infinitely often in any neighborhood. It is not removable, jump, or infinite.

Slant Asymptote

A rational function whose numerator degree exceeds the denominator degree by exactly 1 has an oblique asymptote found by polynomial long division.

End Behavior Growth Order

As x→∞: ln x ≪ x^p (p>0) ≪ a^x (a>1) ≪ x! ≪ x^x. Any quotient across the hierarchy tends to 0 or ∞ accordingly.

Limit of a Composition

lim f(g(x)) = f(lim g(x)) requires f continuous at the inner limit. Without continuity the substitution can fail even when both limits exist.

One-Sided Continuity

f is continuous from the right at a if lim(x→a⁺) f(x)=f(a). Endpoint continuity on a closed interval only requires the one-sided version.

Extreme Value Theorem

A function continuous on a closed bounded interval [a,b] attains both an absolute maximum and an absolute minimum somewhere on that interval.

Removable vs Essential

A discontinuity is removable exactly when the two-sided limit exists but disagrees with (or replaces a missing) function value. Otherwise it is essential.

Piecewise Continuity Parameters

To make a piecewise function continuous at a break x=c, set the two one-sided limits equal to each other and to the assigned value, then solve for the constants.

Vertical Asymptote Test

x=a is a vertical asymptote if at least one one-sided limit is ±∞. A shared factor that cancels completely gives a hole instead.

Indeterminate 1^∞

Rewrite u(x)^v(x) as e^(v ln u). Limits like (1+k/x)^x become e^k because x·ln(1+k/x) → k.

Bounded Times Zero

If |f| ≤ M near a and g→0, then f·g→0 even if f has no limit. This is why x·sin(1/x) → 0 at x=0.

Limits from Tables

A numerical table can suggest but never prove a limit; values may be sampled too coarsely to reveal oscillation or a narrow spike.

Conjugate and Factoring Strategy

For 0/0 forms: factor and cancel for polynomials, multiply by a conjugate for radicals, use a common denominator for compound fractions.

Derivative Hidden as a Limit

A limit of the form lim(h→0) [f(a+h)-f(a)]/h is just f'(a). Recognizing this evaluates hard limits instantly.

Continuity of Elementary Functions

Polynomials, exponentials, sine, and cosine are continuous on all of ℝ; rationals, tangent, and logarithms are continuous on their domains only.

Limits and Absolute Value

|x-a|/(x-a) equals 1 from the right and -1 from the left, so any expression built on it has a jump at a and no two-sided limit.

Infinite Limit vs No Limit

Saying lim f = ∞ is a description of unbounded growth, not an existing limit value; on the AP exam the limit is still said not to exist.

Horizontal Asymptote Crossing

A graph may cross its horizontal asymptote any finite number of times; the asymptote describes only long-run behavior, not a barrier.