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→$\infty$ and/or x→$−\infty$, 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 n$\pm\infty$ is 0; if n=m then the limit is the ratio of leading coefficients; if n>m 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 ($a_n$) and ($b_n$), then ($\lim_{x\to\pm\infty} \frac{p(x)}{q(x)}=\frac{a_n}{b_n}$).

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.