AP Calculus AB Unit 1 Study Guide: Limits, Continuity, Asymptotes, and the IVT

Limit

A description of the value a function f(x) is approaching as x gets close to a particular number (based on nearby behavior, not necessarily the value at the point).

Limit notation

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

Approaching vs. arriving

The idea that limits depend on values of f(x) near x=a, not necessarily on the actual value f(a) (which may be undefined or different).

Two-sided limit

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

Left-hand limit

A one-sided limit that approaches a from values less than a: (\lim_{x\to a^-} f(x)).

Right-hand limit

A one-sided limit that approaches a from values greater than a: (\lim_{x\to a^+} f(x)).

Two-sided limit existence criterion

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

DNE (does not exist) limit

A two-sided limit that fails to exist, commonly because left-hand and right-hand limits do not match (or because behavior is not approaching a single value).

Limit at infinity

A limit describing end behavior as x grows large: (\lim{x\to\infty} f(x)) or (\lim{x\to-\infty} f(x)).

Infinite limit

A limit in which f(x) grows without bound near a point, written as (\lim_{x\to a} f(x)=\infty) or (-\infty).

Direct substitution

Evaluating a limit by plugging in x=a, which works for polynomials and for rational functions when the denominator is not zero at a (i.e., when the function is continuous there).

Continuity (informal)

The idea that a function has no break in its graph near a point (you can draw it without lifting your pencil, locally).

Continuity at a point

f is continuous at x=a if (1) f(a) is defined, (2) (\lim{x\to a} f(x)) exists, and (3) (\lim{x\to a} f(x)=f(a)).

Continuous on an interval [a,b]

Continuous at every interior point (a<x<b), right-continuous at x=a, and left-continuous at x=b.

Right-continuous (at an endpoint)

Continuity condition at the left endpoint x=a of an interval: the right-hand limit matches f(a).

Left-continuous (at an endpoint)

Continuity condition at the right endpoint x=b of an interval: the left-hand limit matches f(b).

Discontinuity

A point x=a where a function is not continuous (at least one of the continuity conditions fails).

Removable discontinuity

A “hole” where the limit exists but f(a) is undefined or f(a) is not equal to the limit; it can be fixed by redefining f(a) to equal the limit.

Jump discontinuity

A discontinuity where both one-sided limits are finite but not equal, so the two-sided limit does not exist.

Infinite (essential) discontinuity

A discontinuity where the function grows without bound near a point (at least one one-sided limit is (\infty) or (-\infty)), typically indicating a vertical asymptote.

Open circle (graph feature)

A marker showing the function is not defined at that exact point (or not taking that y-value there), even though the graph may approach it.

Filled dot (graph feature)

A marker showing the actual function value f(a) at that x-value.

Vertical asymptote

A vertical line x=a where the function is undefined and values of f(x) typically go to (\infty) or (-\infty) as x approaches a.

Horizontal asymptote

A horizontal line y=L that describes end behavior when (\lim{x\to\infty} f(x)=L) or (\lim{x\to-\infty} f(x)=L).

Crossing a horizontal asymptote

The fact that a graph can cross a horizontal asymptote because it describes end behavior, not an absolute boundary.

Limit laws

Algebraic rules that allow splitting and combining limits (sum, difference, constant multiple, product, quotient) when the relevant limits exist and denominators are nonzero.

Sum law for limits

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

Difference law for limits

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

Constant multiple law

If (\lim f(x)=L), then (\lim (c\,f(x))=cL).

Product law for limits

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

Quotient law for limits

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

Indeterminate form (0/0)

A result of direct substitution that signals the need for algebraic simplification; it does not by itself determine the limit.

Factoring and canceling (limit strategy)

A method for (0/0) limits where you factor numerator/denominator and cancel a common factor to remove a hole-causing factor before evaluating the limit.

Common denominator method (limit strategy)

A technique for simplifying complex rational expressions by combining terms into a single fraction to enable cancellation and evaluation.

Conjugate (rationalizing) method

A technique for expressions with radicals: multiply by the conjugate to remove square roots and create a cancelable factor for (0/0) limits.

Difference quotient

The expression (\frac{f(a+h)-f(a)}{h}) used to study rate of change as h→0 (the core setup for derivatives).

Average rate of change

(\frac{f(b)-f(a)}{b-a}), the change in output divided by change in input over an interval.

Secant line

The line through (a,f(a)) and (b,f(b)); its slope equals the average rate of change.

Instantaneous rate of change (preview)

The rate of change at a single point obtained as a limit of average rates of change (letting h→0 in the difference quotient).

Squeeze Theorem

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

Sine bounds inequality

The fact that for all u, (-1\le \sin(u)\le 1), used to squeeze expressions like (x^2\sin(1/x)).

Damped oscillation

Oscillating behavior where the amplitude shrinks to 0 (so a limit can exist even if the function oscillates infinitely often).

Fundamental trig limit

The standard result (\lim_{x\to 0} \frac{\sin x}{x}=1), used to compute many trig limits.

Tangent trig limit

The standard result (\lim_{x\to 0} \frac{\tan x}{x}=1).

Radian measure requirement

The condition that the standard trig limits (like (\sin x/x\to 1)) are valid when angles are measured in radians (as assumed in AP Calculus).

Parameter trig limit (\sin(ax)/x)

The result (\lim_{x\to 0} \frac{\sin(ax)}{x}=a), often found by rewriting as (\frac{\sin(ax)}{ax}\cdot a).

Trig ratio limit (\sin(ax)/\sin(bx))

The result (\lim_{x\to 0} \frac{\sin(ax)}{\sin(bx)}=\frac{a}{b}) (using the (\sin u/u) limit).

Degree comparison rule (rational limits at infinity)

For (\frac{\text{poly degree }n}{\text{poly degree }m}): if nm no finite horizontal asymptote (often unbounded).

Leading coefficient ratio rule

If numerator and denominator have the same degree, (\lim{x\to\infty}\frac{an x^n+\cdots}{bn x^n+\cdots}=\frac{an}{bn}), giving the horizontal asymptote y=(\frac{an}{b_n}).

Intermediate Value Theorem (IVT)

If f is continuous on [a,b] and N is between f(a) and f(b), then there exists c in [a,b] such that f(c)=N (guarantees existence, not the exact value or uniqueness).