AP Calculus AB Unit 1 Notes: Learning Limits (Foundations for Derivatives)

Limit

The value a function’s outputs approach as the inputs get close to a particular number (focuses on nearby behavior, not necessarily the value at the point).

Limit notation

(\lim_{x\to a} f(x)=L), read “the limit of (f(x)) as (x) approaches (a) equals (L).”

Approach (in limits)

Describes getting arbitrarily close to an input value; the function need not be defined or equal to the limit at that exact input.

Two-sided limit

(\lim_{x\to a} f(x)): the behavior of (f(x)) as (x) approaches (a) from both the left and right.

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).

Existence of a two-sided limit

(\lim{x\to a} f(x)) exists iff both one-sided limits exist and are equal: (\lim{x\to a^-} f(x)=\lim_{x\to a^+} f(x)).

DNE (does not exist)

A limit result when the function does not approach a single value (e.g., left/right limits differ, or the function oscillates without settling).

Limit at infinity

(\lim{x\to \infty} f(x)) or (\lim{x\to -\infty} f(x)): describes end behavior for very large positive or negative inputs.

Infinite limit

A limit statement like (\lim_{x\to a} f(x)=\infty), meaning function values grow without bound as (x) approaches (a) (often tied to vertical asymptotes).

Removable discontinuity (hole)

A situation where the graph approaches the same y-value from both sides at (x=a) but has an open circle there; the limit exists even if (f(a)) is missing or different.

Jump discontinuity

A discontinuity where (\lim{x\to a^-} f(x)\neq \lim{x\to a^+} f(x)); the two-sided limit does not exist.

Vertical asymptote behavior

When values blow up near (x=a) (toward (\infty) or (-\infty)), often producing infinite one-sided limits.

Oscillation (near a point)

When a function wiggles infinitely near (x=a) without approaching a single value; the limit does not exist.

Estimating a limit from a graph

A process of tracing the curve toward (x=a) from the left and right to see what y-value(s) the function approaches (separately from any filled dot at (x=a)).

Estimating a limit from a table

Approximating (\lim_{x\to a} f(x)) by checking function values for x-values close to (a) from both sides and seeing whether outputs stabilize (or blow up).

Limit laws

Rules that allow combining limits (sum, difference, constant multiple, product, quotient, power) when the relevant limits exist (and for quotients, the denominator limit is nonzero).

Direct substitution

A method for evaluating limits by plugging in (x=a) when the function is “nice” there (e.g., polynomials, rational functions with nonzero denominator at (a)).

Indeterminate form (0/0)

A result from substitution indicating the expression is undefined at the point and must be simplified to determine the nearby behavior (it does NOT mean the limit is 0).

Factoring and canceling (limit technique)

Simplifying an expression that gives (0/0) by factoring and canceling a common factor (only factors, not terms), then evaluating the remaining limit.

Rationalizing

A technique (often for radicals) where you multiply by a conjugate to remove a square root in the numerator/denominator, enabling cancellation and limit evaluation.

Conjugate

For expressions like (\sqrt{x}-3), the conjugate is (\sqrt{x}+3); multiplying by it uses ((a-b)(a+b)=a^2-b^2) to simplify.

Piecewise function (limits)

A function defined by different formulas on different intervals; limits at a boundary point are found by computing one-sided limits using the applicable formula on each side.

Absolute value one-sided behavior

Near 0, (|x|) equals (x) for (x>0) and (-x) for (x<0), so expressions like (|x|/x) can have different one-sided limits.

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).