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Infinite Limits &
Vertical Asymptotes

Explore how functions behave near values where they blow up to infinity, and learn the deep connection between infinite limits and vertical asymptotes.

AP Calculus BC

What Is an Infinite Limit?

An infinite limit describes the behavior of \( f(x) \) as \( x \) approaches some value \( c \), where the function values grow without bound. We write:

Similarly, \( \lim_{x \to c} f(x) = -\infty \) means the values decrease without bound. Note: the limit does not exist in the traditional sense: "\( = \infty \)" is a description of behavior, not a finite value.

Visualizing Infinite Limits

x y x = c f(x) → +∞ f(x) → +∞

What Is a Vertical Asymptote?

A vertical asymptote is a vertical line \( x = c \) where the graph of a function approaches but never crosses (or touches) in a neighborhood of \( c \), as the function values tend toward \( \pm\infty \).

Definition
The line \( x = c \) is a vertical asymptote of \( f \) if at least one of these is true: \( \lim_{x \to c^+} f(x) = \pm\infty \) or \( \lim_{x \to c^-} f(x) = \pm\infty \).
Key Insight
You only need one one-sided limit to be infinite for \( x = c \) to qualify as a vertical asymptote. Both sides don't have to agree.

The Core Connection

In other words: infinite limits produce vertical asymptotes, and vertical asymptotes are the graphical signature of infinite limits. They are two views of the same phenomenon.

One-Sided Behavior Matters

from left: −∞ from right: +∞ x = c x y

Infinite Limit vs. Limit at Infinity

Infinite Limit Limit at Infinity
Notation \( \lim_{x \to c} f(x) = \pm\infty \) \( \lim_{x \to \infty} f(x) = L \)
What varies? \( x \) approaches a finite value \( c \) \( x \) grows without bound
What blows up? The output \( f(x) \to \pm\infty \) The input \( x \to \pm\infty \)
Graph feature Vertical asymptote Horizontal asymptote
Limit exists? No (DNE in finite sense) Yes, equals \( L \)

Finding VAs in Rational Functions

For \( f(x) = \frac{p(x)}{q(x)} \), vertical asymptotes occur where the denominator is zero but the numerator is not zero (after cancelling common factors).

1
Factor both numerator \( p(x) \) and denominator \( q(x) \) completely.
2
Cancel any common factors. (These give holes, not VAs.)
3
Set the remaining denominator equal to zero: solve \( q(x) = 0 \).
4
Verify each solution by evaluating one-sided limits to confirm the function tends to \( \pm\infty \).

Determining the Sign of Infinity

Once you know \( x = c \) is a VA, determine whether \( f(x) \to +\infty \) or \( -\infty \) on each side using a sign analysis:

1
Pick a test value slightly less than \( c \) (e.g., \( c - 0.01 \)) and slightly greater (e.g., \( c + 0.01 \)).
2
Evaluate the sign of the numerator and denominator at each test value.
3
Determine the sign of the quotient: positive/positive = \( +\infty \), positive/negative = \( -\infty \), etc.

Example 1: \( f(x) = \frac{1}{x - 3} \)

x = 3 1 2 4 x y → −∞ → +∞

Example 2: \( g(x) = \frac{x+1}{(x-2)^2} \)

x = 2 1 3 -1 x y → +∞ → +∞

Example 3: Hole vs. Asymptote

Consider \( h(x) = \frac{x^2 - 4}{x^2 - 5x + 6} \). Factor completely:

Cancel the common factor \( (x - 2) \):

At \( x = 2 \)
Common factor was cancelled, so there is a removable discontinuity (hole), not a VA. The limit exists: \( \lim_{x \to 2} h(x) = \frac{4}{-1} = -4 \).
At \( x = 3 \)
Denominator is zero, numerator \( = 5 \neq 0 \). This is a vertical asymptote. \( \lim_{x \to 3} h(x) = \pm\infty \).

Trig Example: \( \tan x \)

−π/2 π/2 −3π/2 3π/2 x y

Another Source: \( \ln x \)

x = 0 (VA) 1 2 3 x y → −∞

Common Mistakes to Avoid

✕ Forgetting to Cancel
Setting denominator = 0 without factoring first. A common factor yields a hole, not a VA.
✕ Saying the Limit "Equals" Infinity
Writing \( \lim = \infty \) as if \( \infty \) is a real number. The limit technically does not exist; we use \( \infty \) to describe the behavior.
✕ Confusing VA with HA
Vertical asymptotes come from infinite limits (output blows up). Horizontal asymptotes come from limits at infinity (input blows up).

Summary Reference Table

Scenario Behavior Result
\( q(c)=0 \), \( p(c) \neq 0 \) \( f(x) \to \pm\infty \) Vertical asymptote at \( x = c \)
\( q(c)=0 \), \( p(c)=0 \) Factor and cancel Hole (if fully cancelled) or VA (if not)
Even multiplicity in denom. Same sign both sides Both \( +\infty \) or both \( -\infty \)
Odd multiplicity in denom. Opposite signs \( +\infty \) on one side, \( -\infty \) on the other
\( \cos x = 0 \) (for \( \tan x \)) Odd multiplicity VA at \( x = \frac{\pi}{2} + n\pi \)

Question 1 of 5

What is \( \displaystyle\lim_{x \to 5^+} \frac{2}{x - 5} \)?

Question 2 of 5

The function \( f(x) = \frac{x - 1}{(x - 1)(x + 3)} \) has a vertical asymptote at:

Question 3 of 5

If \( \displaystyle\lim_{x \to c^-} f(x) = -\infty \) and \( \displaystyle\lim_{x \to c^+} f(x) = +\infty \), which is true?

Question 4 of 5

How many vertical asymptotes does \( f(x) = \frac{x}{x^2 - 9} \) have?

Question 5 of 5

For \( g(x) = \frac{3}{(x+1)^2} \), what is \( \displaystyle\lim_{x \to -1} g(x) \)?