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

Infinite Limits and Vertical Asymptotes

What an infinite limit means (concept first)

An infinite limit happens when, as you plug x values closer and closer to some number a, the function values grow without bound (either upward or downward). The key idea is that the function is not approaching a finite output—it is “blowing up.”

For example, if as x gets close to a the values of f(x) become arbitrarily large positive numbers, you write:

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

If the values become arbitrarily large negative numbers, you write:

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

This does not mean the limit equals a number called infinity. It means the function grows beyond any finite bound.

Why infinite limits matter

Infinite limits are how calculus makes the idea of a “vertical blow-up” precise. They matter because:

• They are tightly connected to vertical asymptotes, which describe the end behavior of graphs near certain vertical lines.
• They show up constantly with rational functions, logarithms, and some trigonometric functions.
• They help you reason about continuity: if a function has an infinite limit at x=a, it cannot be continuous there (and often isn’t even defined there).

One-sided infinite limits and “which way” the graph blows up

A common AP Calculus situation is that the left-hand and right-hand behaviors near x=a are different. Then you use one-sided limits.

• Left-hand limit:

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

• Right-hand limit:

\lim_{x \to a^+} f(x)

You might see:

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

and

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

In that case, the two-sided limit \lim_{x \to a} f(x) does not exist as a single infinite direction, because the function does not head the same way from both sides.

Vertical asymptotes: what they are and how limits describe them

A vertical asymptote is a vertical line x=a that the graph approaches as x gets close to a (from one side or both), while the function values become unbounded.

The limit language is:

• 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.

Important nuance: many textbooks and AP-style explanations treat “vertical asymptote at x=a” as meaning at least one one-sided limit is infinite. It does not require both sides to go to infinity, and it does not require the function to be undefined at x=a (though that’s the most common situation).

How to find infinite limits for rational functions (the mechanism)

For a rational function

f(x) = \frac{p(x)}{q(x)}

infinite behavior near x=a typically happens when:

• q(a)=0 (denominator is zero), and
• p(a)\neq 0 (numerator is not zero).

Then the fraction’s magnitude tends to blow up because you’re dividing a nonzero number by something approaching 0.

To determine whether the function goes to \infty or -\infty from each side, you do a **sign analysis** near x=a:

1. Factor the denominator (and numerator if needed).
2. Determine the sign of each factor for x just less than a and just greater than a.
3. Combine signs to see whether the fraction is positive or negative while its magnitude grows.

A crucial idea: as x \to a, a factor like x-a changes sign depending on the side:

• If x \to a^-, then x-a is negative.
• If x \to a^+, then x-a is positive.

Removable discontinuities vs vertical asymptotes (what can go wrong)

If both numerator and denominator are zero at x=a, you might have a removable discontinuity (a “hole”) rather than a vertical asymptote. Example structure:

f(x)=\frac{(x-a)g(x)}{(x-a)h(x)}

for x\neq a. Cancelling the common factor can remove the blow-up, producing a finite limit instead. Students often assume “denominator zero means vertical asymptote,” but cancellation changes everything.

Worked Example 1: Identify infinite limits and a vertical asymptote

Consider:

f(x)=\frac{1}{x-2}

As x approaches 2, the denominator approaches 0.

• For x \to 2^-, x-2 is a small negative number, so \frac{1}{x-2} is a large negative number:

\lim_{x \to 2^-} \frac{1}{x-2} = -\infty

• For x \to 2^+, x-2 is a small positive number, so \frac{1}{x-2} is a large positive number:

\lim_{x \to 2^+} \frac{1}{x-2} = \infty

Therefore, x=2 is a vertical asymptote. The two-sided limit does not exist as a single infinity direction because the one-sided limits go to opposite infinities.

Worked Example 2: Distinguish a hole from an asymptote

Consider:

f(x)=\frac{x^2-9}{x-3}

Factor the numerator:

x^2-9=(x-3)(x+3)

So for x\neq 3:

f(x)=\frac{(x-3)(x+3)}{x-3}=x+3

Now the limit is finite:

\lim_{x \to 3} \frac{x^2-9}{x-3} = \lim_{x \to 3} (x+3) = 6

So there is not a vertical asymptote at x=3. Instead, the original expression is undefined at x=3 but approaches 6—a removable discontinuity.

Notation reference (common on AP)

Exam Focus

• Typical question patterns:
  • “Evaluate” an infinite limit such as \lim_{x \to a^-} \frac{p(x)}{q(x)} and state whether it is \infty or -\infty.
  • Given a graph or table, identify where vertical asymptotes occur and match them to infinite one-sided limits.
  • Determine whether a discontinuity is removable (finite limit) or infinite (vertical asymptote).
• Common mistakes:
  • Assuming any denominator zero automatically gives a vertical asymptote (forgetting to check for cancellation).
  • Ignoring one-sided behavior and claiming a two-sided infinite limit exists when the sides go to opposite infinities.
  • Dropping sign analysis—getting \infty vs -\infty wrong.

Limits at Infinity and Horizontal Asymptotes

What “limit at infinity” is really saying

A limit at infinity describes what happens to f(x) when x becomes very large in the positive or negative direction. You are not “plugging in infinity.” Instead, you are looking at end behavior:

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

and

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

If the function approaches a finite number L as x grows without bound, you write:

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

This means you can make f(x) as close to L as you want by choosing x sufficiently large.

Why limits at infinity matter

Limits at infinity help you understand the “big picture” of a function’s graph—especially for rational functions and models. In applications, you often care about long-run behavior:

• In population or economics models, you might ask what value a quantity stabilizes around.
• In physics, a response might approach a steady-state value.

In graphing terms, these limits lead directly to horizontal asymptotes, which describe how the graph behaves far to the left or right.

Horizontal asymptotes: the limit connection

A horizontal asymptote is a horizontal line y=L such that the graph approaches L as x goes to infinity (positive, negative, or both).

Formally:

• If

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

then y=L is a horizontal asymptote (to the right).

• If

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

then y=L is a horizontal asymptote (to the left).

A function can have:

• Two different horizontal asymptotes (one as x \to \infty and another as x \to -\infty).
• No horizontal asymptote.

Also, having a horizontal asymptote does not mean the graph can’t cross it. It often can.

How to compute limits at infinity for rational functions

For rational functions,

f(x)=\frac{p(x)}{q(x)}

the end behavior is determined mainly by the degrees of the polynomials p(x) and q(x).

Let n be the degree of p(x) and m be the degree of q(x).

Case 1: Denominator degree larger (approaches 0)

If n