AP Calculus AB Unit 1 Study Guide: Limits and Continuity (Merged Notes)

Limit

A value that a function’s output approaches as the input gets close to a number, focusing on nearby inputs rather than the value at the point.

Limit notation (two-sided)

(\lim_{x\to a} f(x)=L) means as (x) gets close to (a) from both sides, (f(x)) gets close to (L).

Approaches ((x\to a))

Indicates values of (x) getting close to (a), not necessarily equal to (a).

Function value ((f(a)))

The output of a function at exactly (x=a), if it is defined.

Continuous at a point

A function is continuous at (x=a) if (1) (f(a)) exists, (2) (\lim_{x\to a} f(x)) exists, and (3) the limit equals (f(a)).

Hole (removable discontinuity)

A missing point in an otherwise smooth graph where the limit exists, but the function is undefined or defined to a different value at that input.

Jump discontinuity

A discontinuity where left-hand and right-hand limits both exist but are not equal, so the two-sided limit does not exist.

Infinite (essential) discontinuity

A discontinuity associated with unbounded behavior near a vertical asymptote (the function goes to (\infty) or (-\infty)).

Open circle (graph feature)

A marker showing a point is not included in the graph (often indicating the function is not defined at that (x)-value).

Left-hand limit

(\lim_{x\to a^-} f(x)): the value (f(x)) approaches as (x) approaches (a) from values less than (a).

Right-hand limit

(\lim_{x\to a^+} f(x)): the value (f(x)) approaches as (x) approaches (a) from values greater than (a).

Two-sided limit exists (criterion)

(\lim_{x\to a} f(x)) exists exactly when both one-sided limits exist and are equal.

Indeterminate form (\tfrac{0}{0})

A signal that direct substitution doesn’t determine the limit; the expression must be rewritten/simplified to reveal the limit.

Direct substitution (limit method)

Evaluating a limit by plugging in (x=a); valid when the function is continuous at (a) (e.g., polynomials, rational functions with nonzero denominator).

Polynomial (continuity fact)

A function type that is continuous for all real numbers, so limits can be found by substitution everywhere.

Rational function

A quotient (\frac{p(x)}{q(x)}) of polynomials; continuous wherever (q(x)\neq 0).

Denominator restriction

For a rational function, values that make the denominator 0 are not in the domain and may cause holes or vertical asymptotes.

Limit laws

Rules that allow limits to distribute over algebraic operations (sum, difference, constant multiple, product, quotient, powers) when the needed limits exist.

Sum law (limits)

If (\lim f=L) and (\lim g=M), then (\lim (f+g)=L+M).

Difference law (limits)

If (\lim f=L) and (\lim g=M), then (\lim (f-g)=L-M).

Constant multiple law (limits)

(\lim (c f)=c\lim f=cL) when (\lim f=L).

Product law (limits)

If (\lim f=L) and (\lim g=M), then (\lim (fg)=LM).

Quotient law (limits)

If (\lim f=L) and (\lim g=M) with (M\neq 0), then (\lim \frac{f}{g}=\frac{L}{M}).

Power law (limits)

If (\lim f=L), then (\lim (f(x))^n=L^n) for integer (n).

Factoring and canceling (limit technique)

An algebraic method to remove a common factor causing (\tfrac{0}{0}), often revealing a removable discontinuity (hole).

Canceling factors (restriction)

A common factor can be canceled only when it is a factor of the entire numerator and denominator (not just a term in a sum).

Rationalizing (conjugates)

Multiplying by a conjugate to eliminate radicals and resolve (\tfrac{0}{0}) forms involving square roots.

Conjugate

For (a-\sqrt{b}), the conjugate is (a+\sqrt{b}); their product removes the radical via difference of squares.

Complex fraction

A fraction containing smaller fractions in the numerator and/or denominator; often simplified by combining terms or clearing denominators.

Clear denominators

A technique for simplifying complex fractions by multiplying numerator and denominator by a common denominator.

Special trig limit: (\lim_{x\to 0} \frac{\sin x}{x})

Equals 1 (when angles are in radians); a foundational limit used to evaluate many trig limits.

Special trig limit: (\lim_{x\to 0} \frac{\cos x - 1}{x})

Equals 0 (in radians).

Trig rewrite for (\sin(ax))

(\frac{\sin(ax)}{x}=a\cdot\frac{\sin(ax)}{ax}), which helps apply (\lim_{u\to 0} \frac{\sin u}{u}=1).

Trig ratio limit

(\lim_{x\to 0} \frac{\sin(ax)}{\sin(bx)}=\frac{a}{b}) (constants (a,b), (b\neq 0)).

Radians requirement (trig limits)

The special trigonometric limits are valid when angles are measured in radians.

Piecewise-defined function

A function defined by different formulas on different parts of its domain; limits at boundary points require one-sided analysis.

Boundary point (piecewise)

A value where the rule for a piecewise function changes; evaluate left-hand and right-hand limits using the appropriate formulas.

Domain endpoint one-sided continuity

At a left endpoint, use (\lim{x\to a^+} f(x)=f(a)); at a right endpoint, use (\lim{x\to b^-} f(x)=f(b)).

Infinite limit

A limit statement like (\lim_{x\to a} f(x)=\infty) meaning (f(x)) grows without bound near (a) (not a finite number).

Vertical asymptote

A line (x=a) where the function’s values grow without bound as (x) approaches (a) from at least one side.

Hole vs. vertical asymptote (rational functions)

If a factor causing denominator 0 cancels, it indicates a hole; if it does not cancel, it often indicates a vertical asymptote.

One-sided infinite limits

Limits like (\lim{x\to a^-} f(x)=\infty) and (\lim{x\to a^+} f(x)=-\infty) describing different unbounded behaviors on each side.

Limit at infinity

(\lim_{x\to \infty} f(x)=L) describes end behavior as (x) becomes very large (similarly for (x\to -\infty)).

Horizontal asymptote

A line (y=L) where (f(x)) approaches (L) as (x\to \infty) and/or (x\to -\infty); it describes end behavior and can be crossed.

Degree comparison (rational end behavior)

For (\frac{p(x)}{q(x)}) with degrees (n) and (m): if (n

Leading coefficient ratio rule

When numerator and denominator have the same degree, (\lim_{x\to\infty} \frac{p(x)}{q(x)}) equals (leading coefficient of (p)) / (leading coefficient of (q)).

Composition continuity rule (limits)

If (\lim{x\to a} g(x)=L) and (f) is continuous at (L), then (\lim{x\to a} f(g(x))=f(L)).

Squeeze Theorem

If (g(x)\le f(x)\le h(x)) near (a) and (\lim g=\lim h=L), then (\lim f=L).

Oscillation with shrinking factor

A situation where a wildly oscillating term (like (\sin(1/x))) can still have a limit when multiplied by something that goes to 0 fast enough (e.g., (x^2)).

Intermediate Value Theorem (IVT)

If (f) is continuous on ([a,b]) and (C) lies between (f(a)) and (f(b)), then some (c\in(a,b)) satisfies (f(c)=C); it proves existence, not location or uniqueness.