AP Calculus AB Unit 1 Notes: Learning to Think in Limits

Limit

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

Limit notation

The expression (\lim_{x\to a} f(x)=L), meaning as (x) gets close to (a), (f(x)) gets close to (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 value (f(x)) approaches as (x) approaches (a) using only inputs with (x<a).

Right-hand limit

(\lim_{x\to a^+} f(x)): the value (f(x)) approaches as (x) approaches (a) using only inputs with (x>a).

Existence of a two-sided limit

(\lim{x\to a} f(x)=L) exists iff (\lim{x\to a^-} f(x)=L) and (\lim_{x\to a^+} f(x)=L) (the one-sided limits agree).

Limit vs. function value

A limit depends on values near (a); (f(a)) is the function’s value at (a). The limit can exist even if (f(a)) is undefined or not equal to the limit.

Removable discontinuity (hole)

A point where the graph has an open circle and the function is missing (or redefined) there; the limit may still exist and equals the y-value the curve approaches.

Jump discontinuity

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

Vertical asymptote

A line (x=a) where function values grow without bound (toward (\infty) or (-\infty)) as (x) approaches (a).

Infinite limit

A limit where (f(x)) increases/decreases without bound as (x\to a), written as approaching (\infty) or (-\infty).

DNE (Does Not Exist) for a limit

A limit that is not a single approaching value (commonly because one-sided limits disagree or the function oscillates without settling).

Direct substitution (limit evaluation)

A method to compute a limit by plugging in (x=a) when the function is continuous at (a) (often works for polynomials and many rational functions).

Continuity at a point (limit connection)

If a function is continuous at (x=a), then (\lim_{x\to a} f(x)=f(a)).

Indeterminate form (0/0)

A result from substitution that signals the expression must be simplified (it is not an actual limit value).

Limit laws

Rules that allow combining limits (sum, difference, constant multiple, product, quotient) when the involved limits exist.

Quotient limit law (restriction)

(\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{L}{M}) provided (\lim g(x)=M\neq 0).

Factoring and canceling (limit technique)

An algebra method for (0/0) limits: factor expressions and cancel a common factor to reveal a simpler form valid for (x\neq a) (limits use nearby values).

Rationalizing with conjugates

A method for (0/0) limits with radicals: multiply by a conjugate (multiplying by 1) to eliminate square roots and simplify.

Complex rational expression simplification

A technique for fractions within fractions: rewrite using common denominators to clear the complex fraction, then simplify.

Key trig limit

(\lim_{x\to 0} \frac{\sin x}{x}=1), a foundational result used to evaluate many trigonometric limits.

Trig limit scaling idea

To evaluate (\lim_{x\to 0}\frac{\sin(kx)}{x}), rewrite as (k\cdot\frac{\sin(kx)}{kx}) so the limit becomes (k).

Oscillation causing a limit to fail

A situation like (\sin(1/x)) as (x\to 0), where outputs keep varying between values and do not settle to one number (so the limit DNE).

Estimating limits from graphs and tables

A method to infer (\lim_{x\to a} f(x)) by checking values as (x) approaches (a) from the left and right and seeing what output value the function approaches.

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).