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y = L

Limits at Infinity &
Horizontal Asymptotes

Explore what happens to functions as \( x \to \pm\infty \) and discover the connection between end behavior and horizontal asymptotes.

AP Calculus AB

What Is a Limit at Infinity?

A limit at infinity describes the value that \( f(x) \) approaches as \( x \) grows without bound. We write:

This means that as \( x \) increases (or decreases) without bound, the output values of \( f(x) \) get arbitrarily close to the finite number \( L \).

Visualizing End Behavior

x y y = 2 As x grows, f(x) approaches 2 1 5 10 2 5 0

What Is a Horizontal Asymptote?

A horizontal asymptote is a horizontal line \( y = L \) that the graph of \( f(x) \) approaches as \( x \to \infty \) or \( x \to -\infty \).

A function can have at most two horizontal asymptotes: one for each direction. It can also have zero or one.

The Core Connection

Limit at Infinity
\( \displaystyle \lim_{x \to \infty} f(x) = L \) tells us the value the function approaches.
Horizontal Asymptote
The line \( y = L \) is the geometric representation of that limiting value on the graph.

Theorem: Rational Functions

For a rational function \( \displaystyle f(x) = \frac{a_n x^n + \cdots}{b_m x^m + \cdots} \), the limit at infinity depends on comparing the degrees \( n \) and \( m \):

1
If \( n < m \): \( \displaystyle \lim_{x \to \pm\infty} f(x) = 0 \). Horizontal asymptote at \( y = 0 \).
2
If \( n = m \): \( \displaystyle \lim_{x \to \pm\infty} f(x) = \frac{a_n}{b_m} \). Horizontal asymptote at \( y = \frac{a_n}{b_m} \).
3
If \( n > m \): The limit is \( \pm\infty \). No horizontal asymptote.

Three Cases Visualized

\( n < m \)
HA: \( y = 0 \)
\( n = m \)
HA: \( y = \frac{a}{b} \)
\( n > m \)
 No HA Degree num < denom Degree num = denom Degree num > denom

Horizontal vs Vertical Asymptotes

Horizontal Asymptote Vertical Asymptote
Form \( y = L \) \( x = a \)
Found by \( \lim_{x \to \pm\infty} f(x) \) \( \lim_{x \to a} f(x) = \pm\infty \)
Can be crossed? Yes No
Max count At most 2 Unlimited
Describes End behavior Local blow-up

How to Find Horizontal Asymptotes

1
Identify the function type. Is it rational, exponential, or something else?
2
For rational functions: compare the degree of the numerator \( n \) to the degree of the denominator \( m \).
3
Divide every term by the highest power of \( x \) in the denominator. Simplify.
4
Evaluate the limit as \( x \to \infty \) and as \( x \to -\infty \). If the result is a finite number \( L \), then \( y = L \) is a horizontal asymptote.
5
State both limits. They may differ: you could have \( y = 2 \) as \( x \to \infty \) and \( y = -1 \) as \( x \to -\infty \).

Example 1: Equal Degrees

Find the horizontal asymptote of \( \displaystyle f(x) = \frac{3x^2 - 5}{2x^2 + 1} \).

1
Degrees: numerator degree = 2, denominator degree = 2. They are equal.
2
Divide by \( x^2 \): \( \displaystyle \frac{3 - \frac{5}{x^2}}{2 + \frac{1}{x^2}} \)
3
Take the limit: As \( x \to \infty \), the \( \frac{5}{x^2} \) and \( \frac{1}{x^2} \) terms vanish, giving \( \displaystyle \frac{3}{2} \).

Example 2: Numerator Smaller

Find the horizontal asymptote of \( \displaystyle g(x) = \frac{4x + 1}{x^3 - 2} \).

1
Degrees: numerator = 1, denominator = 3. Since \( 1 < 3 \), the limit is 0.
2
Divide by \( x^3 \): \( \displaystyle \frac{\frac{4}{x^2} + \frac{1}{x^3}}{1 - \frac{2}{x^3}} \to \frac{0}{1} = 0 \)

Example 3: No Horizontal Asymptote

Evaluate \( \displaystyle \lim_{x \to \infty} \frac{x^3 + 2x}{5x - 3} \). Does a horizontal asymptote exist?

1
Degrees: numerator = 3, denominator = 1. Since \( 3 > 1 \), the limit is not finite.
2
Divide by \( x \): \( \displaystyle \frac{x^2 + 2}{5 - \frac{3}{x}} \to \frac{\infty}{5} = \infty \)

Beyond Rational: Exponential Decay

Not all functions with horizontal asymptotes are rational. Consider \( \displaystyle h(x) = 5 - 3e^{-x} \).

x y y = 5 5 2 0 0 3 6

Two Different Asymptotes

Some functions approach different values at \( +\infty \) and \( -\infty \). Consider \( \displaystyle f(x) = \frac{2x}{\sqrt{x^2 + 1}} \).

As \( x \to +\infty \)
\( \sqrt{x^2+1} \approx x \), so \( f(x) \to \frac{2x}{x} = 2 \). HA: \( y = 2 \).
As \( x \to -\infty \)
\( \sqrt{x^2+1} \approx |x| = -x \), so \( f(x) \to \frac{2x}{-x} = -2 \). HA: \( y = -2 \).

Common Mistakes

✕ Ignoring the Sign of \( x \)
When dividing by \( x \) under a square root, students forget that \( x < 0 \) for \( x \to -\infty \). This flips the sign of the limit.
✕ Plugging in \( \infty \)
Writing \( \frac{\infty}{\infty} = 1 \) is invalid. You must simplify algebraically first, then evaluate the limit.
✕ Assuming HA Cannot Be Crossed
Unlike vertical asymptotes, a function can cross its horizontal asymptote at finite \( x \)-values. The HA describes only end behavior.

Summary of Key Results

Function Type Limit as \( x \to \infty \) Horizontal Asymptote
\( \frac{a}{x^n} \)   (\( n > 0 \)) 0 \( y = 0 \)
\( \frac{ax^n + \cdots}{bx^n + \cdots} \) \( \frac{a}{b} \) \( y = \frac{a}{b} \)
\( \frac{ax^n + \cdots}{bx^m + \cdots} \), \( n < m \) 0 \( y = 0 \)
\( \frac{ax^n + \cdots}{bx^m + \cdots} \), \( n > m \) \( \pm\infty \) None
\( c + ke^{-x} \) \( c \) \( y = c \)

Question 1 of 5

What is \( \displaystyle \lim_{x \to \infty} \frac{5x^2 + 3}{2x^2 - 7} \)?

Question 2 of 5

The function \( \displaystyle g(x) = \frac{3x}{x^2 + 4} \) has a horizontal asymptote at:

Question 3 of 5

Which statement is true about horizontal asymptotes?

Question 4 of 5

What is \( \displaystyle \lim_{x \to -\infty} \frac{2x}{\sqrt{x^2 + 1}} \)?

Question 5 of 5

Which function has no horizontal asymptote?