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

More AP Calculus AB guides

• Derivatives
• Applications of Derivatives
• Integration
• Differential Equations

Limits and Continuity

Limit: value function approaches as input approaches some value. Continuity: function is unbroken at a point; requires limit exists and equals function value.

Limit Definition

lim(x→a) f(x) = L means for every ε > 0, there exists δ > 0 such that |x-a| < δ implies |f(x)-L| < ε; rigorous definition of limit.

One-Sided Limits

Left limit: lim(x→a⁻) f(x); right limit: lim(x→a⁺) f(x). Two-sided limit exists only if both one-sided limits exist and are equal.

Limit Laws

Sum, product, quotient, power rules; if lim(x→a) f(x) = L and lim(x→a) g(x) = M, then lim(x→a) [f(x)+g(x)] = L+M (and similar for other operations).

Infinite Limits

lim(x→a) f(x) = ∞ means function grows unbounded; often indicates vertical asymptote at x = a.

Limits at Infinity

lim(x→∞) f(x) = L means function approaches L as x grows; describes horizontal asymptotes; key for analyzing end behavior.

Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x) near a, and lim(x→a) g(x) = lim(x→a) h(x) = L, then lim(x→a) f(x) = L; useful for hard-to-evaluate limits.

Continuity at a Point

f is continuous at a if lim(x→a) f(x) = f(a); three conditions: limit exists, function defined at a, and they're equal.

Types of Discontinuities

Removable: hole (limit exists, f(a) undefined or wrong); jump: left/right limits differ; infinite: vertical asymptote.

Intermediate Value Theorem

If f is continuous on [a,b] and N between f(a) and f(b), then there exists c in (a,b) where f(c) = N; guarantees solutions exist.

Asymptotes

Vertical: x = a where function undefined/unbounded (lim = ±∞); horizontal: y = L where lim(x→±∞) f(x) = L; oblique: linear asymptote.

Polynomials and Limits

Polynomial limits: substitute directly (no indeterminate form). Rational functions: factor and cancel before substituting to resolve 0/0 form.

Indeterminate Forms

0/0, ∞/∞, 0·∞, ∞-∞, 0⁰, 1^∞, ∞⁰: these are ambiguous; require algebraic manipulation or L'Hôpital's rule to evaluate.

L'Hôpital's Rule (Preview)

For indeterminate forms 0/0 or ∞/∞: lim f(x)/g(x) = lim f'(x)/g'(x); covered in detail in Unit 5.

Trigonometric Limits

Key limit: lim(x→0) sin(x)/x = 1; useful for evaluating trig limit problems; derived using squeeze theorem.

Exponential and Logarithmic Limits

lim(x→∞) (1+1/x)^x = e; lim(x→0⁺) x·ln(x) = 0; key limits for transcendental functions.

Continuity of Basic Functions

Polynomials, rational functions (where defined), trig functions, exponential, logarithm are continuous on their domains.

Composition Continuity

If f continuous at g(a) and g continuous at a, then (f∘g) continuous at a; allows analyzing complex composite functions.

Extreme Value Theorem

If f continuous on [a,b], then f attains max and min on [a,b]; guarantees extrema exist for continuous functions on closed intervals.

Informal Limit Idea

Limit describes behavior of function near (not at) a point; foundation for calculus; different from value of function.

Conjugate Technique

For limits with a radical difference such as (√(x+9)-3)/x, multiply numerator and denominator by the conjugate to convert a 0/0 form into a cancelable rational expression.

Rationalizing at Infinity

For √(ax²+bx) - cx as x→∞, multiply by the conjugate over itself; the x² terms cancel and the surviving linear terms give a finite limit.

Half-Angle Limit

lim(x→0) (1-cos x)/x² = 1/2, obtained by multiplying by (1+cos x)/(1+cos x) and using sin²x/x² → 1.

Bounded Times Vanishing

If |g(x)| ≤ M near a and f(x)→0, then f(x)g(x)→0. This resolves oscillating products such as x·sin(1/x) at x = 0.

Continuous but Not Differentiable

x·sin(1/x) extended by 0 at the origin is continuous there, yet the difference quotient sin(1/h) oscillates, so no derivative exists.

Sign Analysis of Infinite Limits

For a one-sided infinite limit, test the sign of numerator and denominator separately just to one side of the value; a squared denominator keeps a fixed sign from both sides.

Degree Comparison at Infinity

For rational functions: lower numerator degree gives limit 0, equal degrees give the ratio of leading coefficients, higher numerator degree gives ±∞.

Two Horizontal Asymptotes

Functions like 2x/√(x²+1) approach 2 as x→∞ and -2 as x→-∞, because √(x²) = |x| forces a sign change on the negative side.

Indeterminate Powers

Forms 0⁰, 1^∞, and ∞⁰ are handled by writing y = f(x)^g(x), taking ln to get g(x)·ln f(x), evaluating that limit, then exponentiating.

Growth Rate Hierarchy

As x→∞, ln x grows slower than any positive power of x, which grows slower than any exponential a^x with a > 1.

Existence vs Value

A limit can exist where the function is undefined, and a function can be defined where its limit fails to exist; the two ideas are logically independent.

Squeeze with Unknown Function

If bounding functions agree in limit at a point, the trapped function inherits that limit even when no formula for it is available.

Root Existence Argument

To justify a root, define g as the difference of the two sides, confirm continuity, and exhibit a sign change; IVT then supplies the root.

Continuity of Piecewise Functions

Continuity at a seam x = c requires the two branch expressions to agree at c; solving that equation determines unknown parameters.

Removable Discontinuity in Trig Quotients

sin(x)/(x(x-π)) has a removable discontinuity at x = 0 because sin(x)/x → 1, leaving the finite value -1/π.

Limits of Composite Functions

lim f(g(x)) equals f(lim g(x)) only when f is continuous at that inner limit; otherwise evaluate the inner behavior and read f from one side.

Vertical Asymptote vs Hole

After full factoring, a factor that cancels gives a hole; a factor remaining in the denominator gives a vertical asymptote.

Limits from Tables

A table can suggest but never prove a limit; values can be chosen to hide oscillation, so tabular evidence must be supported by algebra.

Difference Quotient Limits

Limits of the form lim(h→0)[f(a+h)-f(a)]/h that look like hard limit problems are really derivative evaluations in disguise.

Slant Asymptote

When the numerator degree exceeds the denominator degree by exactly one, long division gives a linear quotient that the graph approaches at ±∞.