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\( y=L \)

Limits at Infinity &
Horizontal Asymptotes

Explore what happens to \( f(x) \) as \( x \to \pm\infty \) and how this determines the end behavior of rational functions.

AP Calculus BC

What Is a Limit at Infinity?

A limit at infinity describes the value that \( f(x) \) approaches as the input \( x \) grows without bound in the positive or negative direction.

This is different from an ordinary limit at a point: we are not approaching a finite \( x \)-value, but instead asking about the long-run trend of the function.

Visualizing End Behavior

x y \( y = 2 \) f(x) approaches 2 f(x) approaches 2 -2 -1 1 2

Why Does End Behavior Matter?

Curve Sketching
Horizontal asymptotes anchor the far-left and far-right portions of a graph, giving you the overall shape.
Real-World Models
Population models, drug concentrations, and economics use limits at infinity to describe long-term equilibrium values.
AP Exam Staple
Both the AP Calculus AB and BC exams test horizontal asymptotes via limits, graph analysis, and free-response questions.

The Formal Definition

A function can have at most two horizontal asymptotes: one as \( x \to \infty \) and a possibly different one as \( x \to -\infty \). Some functions have the same asymptote in both directions, and some have none at all.

The Degree Comparison Theorem

For rational functions \( f(x) = \dfrac{a_n x^n + \cdots}{b_m x^m + \cdots} \), compare the degree of the numerator (\( n \)) with the degree of the denominator (\( m \)):

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

Three Cases Visualized

\( n < m \) \( y=0 \) \( n = m \) \( y=3 \) \( n > m \) No HA HA: \( y=0 \) HA: \( y=\frac{a_n}{b_m} \) No HA

Horizontal vs. Vertical Asymptotes

Horizontal Vertical
Form \( y = L \) \( x = a \)
Found by \( \lim_{x \to \pm\infty} f(x) \) \( \lim_{x \to a} f(x) = \pm\infty \)
Max count At most 2 Unlimited
Graph crosses? Yes, may cross Never crosses
Describes End behavior Local blowup

The Divide-by-Highest-Power Method

For any rational function, you can find the limit at infinity algebraically by dividing every term by the highest power of \( x \) in the denominator.

This technique works universally and avoids the need to memorize cases: just divide and simplify.

Step-by-Step Approach

1
Identify the type: Is \( f(x) \) a rational function, a radical expression, or another form?
2
Compare degrees of numerator and denominator. If equal or numerator is smaller, an HA exists.
3
Divide every term by the highest power of \( x \) in the denominator.
4
Evaluate the limit: Replace \( \frac{1}{x^k} \to 0 \). The remaining constant is \( L \).
5
State the asymptote: Write \( y = L \). Check both \( x \to +\infty \) and \( x \to -\infty \).

Example 1: Equal Degrees

1
Degrees: Numerator degree = 3, denominator degree = 3. They are equal.
2
Divide by \( x^3 \): \(\dfrac{5 - \frac{2}{x^2}}{3 + \frac{7}{x^3}}\)
3
Take the limit: As \( x \to \infty \), the \( \frac{1}{x^k} \) terms vanish: \(\dfrac{5 - 0}{3 + 0} = \dfrac{5}{3}\)

Example 2: Numerator Degree Smaller

1
Degrees: Numerator = 1, denominator = 2. So \( n < m \).
2
Divide by \( x^2 \): \(\dfrac{\frac{4}{x} + \frac{1}{x^2}}{1 - \frac{9}{x^2}}\)
3
Take the limit: \(\dfrac{0 + 0}{1 - 0} = 0\)

Example 3: Two Different Asymptotes

As \( x \to +\infty \): Since \( x > 0 \), we have \( \sqrt{x^2} = x \). Divide numerator and denominator by \( x \):

As \( x \to -\infty \): Since \( x < 0 \), we have \( \sqrt{x^2} = -x \). The sign flips:

This function has two horizontal asymptotes: \( y = 3 \) and \( y = -3 \).

Beyond Rational Functions

The concept of limits at infinity applies to all functions, not just rational ones. Some important non-rational examples:

\( e^{-x} \)
\(\lim_{x \to \infty} e^{-x} = 0\). Exponential decay approaches \( y = 0 \).
\( \arctan(x) \)
\(\lim_{x \to \infty} \arctan(x) = \frac{\pi}{2}\) and \(\lim_{x \to -\infty} \arctan(x) = -\frac{\pi}{2}\). Two HAs.
\( \sin(x)/x \)
\(\lim_{x \to \infty} \frac{\sin x}{x} = 0\) by the Squeeze Theorem since \( |\sin x| \leq 1 \).

Common Mistakes

✕ Ignoring the sign of \( x \)
When \( x \to -\infty \), remember \( \sqrt{x^2} = |x| = -x \), not \( x \). This flips the sign and can produce a different asymptote.
✕ Confusing HA with VA
A horizontal asymptote is about \( x \to \pm\infty \). A vertical asymptote is about \( f(x) \to \pm\infty \) at a finite \( x \)-value. Don't mix the two.
✕ Saying "HA can't be crossed"
Functions can cross horizontal asymptotes at finite \( x \)-values. For example, \( \frac{\sin x}{x} \) crosses \( y = 0 \) infinitely often.

Summary of Key Rules

Condition Limit Horizontal Asymptote
\( \deg(P) < \deg(Q) \) \( 0 \) \( y = 0 \)
\( \deg(P) = \deg(Q) \) \( \frac{a_n}{b_m} \) \( y = \frac{a_n}{b_m} \)
\( \deg(P) > \deg(Q) \) \( \pm\infty \) None
Radical with \( x \to -\infty \) Check sign of \( x \) May differ from \( +\infty \)
\( e^{-x} \), \( \arctan(x) \) Use known limits \( y = 0 \), \( y = \pm\frac{\pi}{2} \)

Question 1 of 5

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

Question 2 of 5

What is the horizontal asymptote of \( f(x) = \dfrac{2x}{x^3 + 1} \)?

Question 3 of 5

Which statement about horizontal asymptotes is true?

Question 4 of 5

What is \(\displaystyle\lim_{x \to -\infty} \frac{5x}{\sqrt{x^2 + 9}}\)?

Question 5 of 5

If \(\displaystyle\lim_{x \to \infty} f(x) = 4\) and \(\displaystyle\lim_{x \to -\infty} f(x) = -1\), how many horizontal asymptotes does \( f \) have?