AP Calculus AB Unit 1 Notes: Infinity in Limits, Asymptotes, and Existence Theorems

Infinite limit

A limit where f(x) grows without bound (toward positive or negative infinity) as x approaches a finite value a, rather than approaching a finite number.

(\lim_{x\to a} f(x)=\infty)

As x approaches a, the function values become arbitrarily large positive numbers (increase without bound).

(\lim_{x\to a} f(x)=-\infty)

As x approaches a, the function values become arbitrarily large negative numbers (decrease without bound).

One-sided limit

A limit that describes function behavior as x approaches a from only one side (left or right).

Left-hand limit (\lim_{x\to a^-} f(x))

The limit of f(x) as x approaches a using x-values less than a (approaching from the left).

Right-hand limit (\lim_{x\to a^+} f(x))

The limit of f(x) as x approaches a using x-values greater than a (approaching from the right).

Two-sided limit does not exist (DNE) due to opposite infinities

If (\lim{x\to a^-} f(x)=\infty) and (\lim{x\to a^+} f(x)=-\infty) (or vice versa), then (\lim_{x\to a} f(x)) does not exist as a single limit value/direction.

Vertical asymptote

A vertical line x=a that the graph approaches as x gets close to a (from one side or both) while f(x) becomes unbounded (goes to (\pm\infty)).

Vertical asymptote test (limit form)

If (\lim{x\to a^-} f(x)=\pm\infty) or (\lim{x\to a^+} f(x)=\pm\infty), then x=a is a vertical asymptote.

Rational function

A function of the form (f(x)=\frac{p(x)}{q(x)}), where p(x) and q(x) are polynomials.

Infinite behavior in rational functions (typical condition)

For (f(x)=\frac{p(x)}{q(x)}), an infinite limit near x=a typically occurs when (q(a)=0) and (p(a)\neq 0) (nonzero divided by something approaching 0).

Sign analysis (near a vertical asymptote)

A method to determine whether a rational function approaches (\infty) or (-\infty) from each side of x=a by factoring and checking the sign of factors just left and right of a.

Factor sign change for (x-a)

As x approaches a from the left, (x-a

Removable discontinuity

A discontinuity where the function is undefined at x=a but the limit as x→a is finite (often shown as a “hole” that can be removed by simplifying).

Hole (in a graph)

A missing point caused by a removable discontinuity; the function may approach a finite value there even though it is undefined at that x-value.

Cancellation (common factor)

Simplifying a rational expression by canceling a shared factor (like (x−a)) in numerator and denominator, which can eliminate an apparent vertical asymptote and reveal a removable discontinuity.

Example: (\frac{1}{x-2}) near x=2

As x→2−, the function → (-\infty); as x→2+, the function → (\infty). Therefore x=2 is a vertical asymptote and the two-sided limit DNE.

Example: (\frac{x^2-9}{x-3}) at x=3

Factoring gives (\frac{(x-3)(x+3)}{x-3}=x+3) for x≠3, so (\lim_{x\to 3} \frac{x^2-9}{x-3}=6); this is a removable discontinuity (not a vertical asymptote).

Limit at infinity

A limit describing end behavior as x becomes very large positive or very large negative (x→∞ or x→−∞), not by “plugging in” infinity but by analyzing long-run behavior.

Horizontal asymptote

A horizontal line y=L that the graph approaches as x→∞ and/or x→−∞, corresponding to (\lim{x\to\infty} f(x)=L) and/or (\lim{x\to-\infty} f(x)=L).

Horizontal asymptote is not a barrier

Having a horizontal asymptote y=L does not prevent the graph from crossing y=L; it only describes end behavior.

Degree comparison (rational limits at infinity)

For (\frac{p(x)}{q(x)}) with degrees n and m: if nm then there is no horizontal asymptote (the function does not approach a finite constant).

Ratio of leading coefficients (same degree case)

If p and q have the same degree n with leading coefficients (an) and (bn), then (\lim{x\to\pm\infty} \frac{p(x)}{q(x)}=\frac{an}{b_n}).

Divide by highest power technique

A method for rational limits at infinity: divide numerator and denominator by the highest power of x to make dominant terms visible and use that terms like 1/x and 1/x² go to 0 as x→∞.

Intermediate Value Theorem (IVT)

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