AP Calculus BC Unit 1 (Limits and Continuity): Comprehensive Study Notes

Limit

A value that a function’s output approaches as the input gets close to a particular number (describes behavior near a point, not necessarily at the point).

Limit notation

The statement (\lim_{x\to a} f(x)=L), meaning as (x) approaches (a), (f(x)) approaches (L).

Two-sided limit

A limit (\lim_{x\to a} f(x)) that considers approaching (a) from both the left and the right.

Left-hand limit

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

Right-hand limit

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

Limit exists (two-sided)

A two-sided limit exists exactly when both one-sided limits exist and are equal.

Function value

The actual output (f(a)) at the input (a), if the function is defined there.

Indeterminate form (0/0)

A result from direct substitution that signals the expression must be rewritten/simplified; it does not automatically mean the limit is 0 or undefined.

Direct substitution

Evaluating a limit by plugging in the approaching input value, which works for well-behaved (continuous) functions like polynomials.

Limit laws

Algebraic rules that allow limits to be split across operations (sum, product, quotient) when the relevant limits exist.

Sum law (limits)

If (\lim f(x)) and (\lim g(x)) exist, then (\lim (f(x)+g(x)) = \lim f(x)+\lim g(x)).

Product law (limits)

If (\lim f(x)) and (\lim g(x)) exist, then (\lim (f(x)g(x)) = (\lim f(x))(\lim g(x))).

Quotient law (limits)

If (\lim f(x)) and (\lim g(x)) exist and (\lim g(x)\neq 0), then (\lim \frac{f(x)}{g(x)}=\frac{\lim f(x)}{\lim g(x)}).

Removable discontinuity

A discontinuity where the limit exists but the function value is missing or does not match the limit (graphically appears as a hole).

Hole (open circle)

Graph feature indicating the function is not defined at that point (or is defined differently), though the limit there may still exist.

Factoring (for limits)

Rewriting expressions (often polynomials) as products to reveal cancelable factors and resolve indeterminate forms like (0/0).

Canceling a common factor

After factoring, removing a shared factor in numerator and denominator (valid only as a factor, not across addition), often revealing a removable discontinuity.

Rationalizing

A technique for limits with radicals that multiplies by a conjugate to eliminate radicals and simplify the expression.

Conjugate

For an expression (a-b), the conjugate is (a+b); multiplying them uses ((a-b)(a+b)=a^2-b^2).

Piecewise-defined function

A function defined by different formulas on different intervals; limits at boundary points require one-sided analysis using the correct formula on each side.

Boundary point (piecewise)

An input where a piecewise rule changes; the two-sided limit depends on matching the left-hand and right-hand limits.

Foundational trig limit (\lim_{x\to 0} \frac{\sin x}{x}=1)

A key trigonometric limit used throughout calculus (valid when angles are in radians).

Radians (in trig limits)

The required angle unit for standard trig limits like (\lim_{x\to 0} \frac{\sin x}{x}=1) to be true as stated.

Limit (\lim_{x\to 0}\frac{1-\cos x}{x^2}=\frac12)

A standard trigonometric limit often derived using conjugates and (1-\cos^2 x=\sin^2 x).

Pattern (\lim_{x\to 0}\frac{\sin(ax)}{x}=a)

A trig-limit pattern found by rewriting as (a\cdot \frac{\sin(ax)}{ax}).

Pattern (\lim_{x\to 0}\frac{\sin(ax)}{\sin(bx)}=\frac{a}{b})

A trig-limit pattern using sine-over-angle reasoning for both numerator and denominator.

Squeeze Theorem

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

Oscillation (in limits)

Behavior where a function keeps fluctuating (e.g., sine terms); if bounded, it may be handled using the Squeeze Theorem.

Infinite limit

A limit where function outputs grow without bound near an input, written like (\lim_{x\to a} f(x)=\infty) or (-\infty).

Vertical asymptote

A vertical line (x=a) where a function is undefined or becomes unbounded; often detected via infinite one-sided limits.

One-sided infinite limits

Limits near a vertical asymptote that can differ on each side (e.g., one side (\infty), the other (-\infty)).

Limit at infinity

A statement about end behavior as inputs grow without bound, such as (\lim_{x\to\infty} f(x)=L).

End behavior

How a function behaves as (x\to\infty) or (x\to -\infty).

Horizontal asymptote

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

Leading coefficient ratio (equal degrees)

For a rational function with equal numerator/denominator degrees, the limit at infinity equals the ratio of leading coefficients.

Degree (of a polynomial)

The highest power of (x) in the polynomial; used to analyze limits at infinity for rational functions.

Rational function

A function of the form (\frac{P(x)}{Q(x)}) where (P) and (Q) are polynomials.

Rational limit at infinity: numerator degree < denominator degree

If the denominator has higher degree, (\lim_{x\to\pm\infty}\frac{P(x)}{Q(x)}=0) (horizontal asymptote (y=0)).

Rational limit at infinity: equal degrees

If degrees match, (\lim_{x\to\pm\infty}\frac{P(x)}{Q(x)}) equals the ratio of leading coefficients.

Rational limit at infinity: numerator degree > denominator degree

If the numerator has higher degree, the function does not approach a finite horizontal asymptote (the limit at infinity is unbounded/diverges).

Continuity

The property of having no breaks in the graph; many calculus theorems and techniques rely on it.

Continuity at a point (3 conditions)

At (x=a): (1) (f(a)) exists, (2) (\lim{x\to a} f(x)) exists, and (3) (\lim{x\to a} f(x)=f(a)).

Continuity on an interval

A function is continuous on an interval if it is continuous at every point in that interval.

Endpoint continuity (one-sided)

On a closed interval, continuity at an endpoint is checked using the appropriate one-sided limit from within the interval.

Jump discontinuity

A discontinuity where the left-hand and right-hand limits both exist and are finite, but they are not equal.

Infinite (essential) discontinuity

A discontinuity caused by unbounded behavior near a point (typically associated with a vertical asymptote).

Redefining a function to remove a discontinuity

Making a function continuous at a removable discontinuity by defining the missing value to equal the limit at that input.

Intermediate Value Theorem (IVT)

If (f) is continuous on ([a,b]) and (N) is between (f(a)) and (f(b)), then some (c\in[a,b]) satisfies (f(c)=N).

IVT (existence guarantee)

IVT guarantees at least one solution exists in the interval but does not tell where it is or how many solutions there are.

Limit vs. function value

The limit describes what (f(x)) approaches near (a), while the function value is (f(a)); they can be different (e.g., a hole with a filled dot elsewhere).