Limits at Infinity – AP Calculus

So far, we've looked at limits as $x$ approaches a finite number. But what happens to a function as $x$ grows larger and larger — heading toward infinity? That's called a limit at infinity, and it tells us the function's end behavior (what the graph does far to the right or far to the left).

\lim_{x \to \infty} f(x) = L \quad \text{means: as $x$ increases without bound, $f(x)$ gets arbitrarily close to $L$} $$ $$ \lim_{x \to -\infty} f(x) = L \quad \text{means: as $x$ decreases without bound, $f(x)$ gets arbitrarily close to $L$}

📌 Horizontal Asymptotes

If $\displaystyle \lim_{x \to \infty} f(x) = L$ or $\displaystyle \lim_{x \to -\infty} f(x) = L$, then the line $y = L$ is a horizontal asymptote of the graph of $f$.

📊 Visualizing Limits at Infinity

Use the interactive graph below. Move the slider to see how each function behaves as $x$ moves far to the right or left:

🧮 How to Compute Limits at Infinity

The most common types of limits at infinity come from rational functions (ratios of polynomials). Here's the rule:

📐 Rule for Rational Functions

For $f(x) = \frac{a_n x^n + \dots}{b_m x^m + \dots}$ as $x \to \pm\infty$:

If $n < m$ (degree of numerator < degree of denominator), $\displaystyle \lim_{x \to \pm\infty} f(x) = 0$
If $n = m$ (degrees equal), $\displaystyle \lim_{x \to \pm\infty} f(x) = \frac{a_n}{b_m}$ (ratio of leading coefficients)
If $n > m$ (degree of numerator > degree of denominator), the limit is $\infty$ or $-\infty$ (or does not exist)

Example 1Degree of Denominator Larger (Limit = 0)

Find $\displaystyle \lim_{x \to \infty} \frac{5x + 2}{x^2 - 3}$.

Degree of numerator: $1$
Degree of denominator: $2$
Since $1 < 2$, $\displaystyle \lim_{x \to \infty} \frac{5x + 2}{x^2 - 3} = 0$

As $x$ grows huge, the denominator grows much faster than the numerator, so the fraction shrinks to $0$.

Example 2Degrees Equal (Limit = Ratio of Leading Coefficients)

Find $\displaystyle \lim_{x \to \infty} \frac{3x^2 + 2x - 1}{x^2 + 5}$.

Degree of numerator: $2$
Degree of denominator: $2$
Leading coefficient ratio: $\frac{3}{1} = 3$
$\displaystyle \lim_{x \to \infty} \frac{3x^2 + 2x - 1}{x^2 + 5} = 3$

Example 3Degree of Numerator Larger (Limit = ∞)

Find $\displaystyle \lim_{x \to \infty} \frac{x^3 - x}{x^2 + 1}$.

Degree of numerator: $3$
Degree of denominator: $2$
Since $3 > 2$, $\displaystyle \lim_{x \to \infty} \frac{x^3 - x}{x^2 + 1} = \infty$

The numerator grows faster, so the function grows without bound.

Example 4Negative Infinity

Find $\displaystyle \lim_{x \to -\infty} \frac{x^3 - x}{x^2 + 1}$.

Same degree comparison: $3 > 2$
But as $x \to -\infty$, $x^3$ becomes very negative (since odd power)
$\displaystyle \lim_{x \to -\infty} \frac{x^3 - x}{x^2 + 1} = -\infty$

Example 5Method: Divide by Highest Power

Find $\displaystyle \lim_{x \to \infty} \frac{2x^2 + 3x}{x^2 - 5}$ by dividing numerator and denominator by $x^2$.

$\displaystyle \frac{2x^2 + 3x}{x^2 - 5} = \frac{2 + \frac{3}{x}}{1 - \frac{5}{x^2}}$
As $x \to \infty$, $\frac{3}{x} \to 0$ and $\frac{5}{x^2} \to 0$
So $\displaystyle \lim_{x \to \infty} \frac{2 + \frac{3}{x}}{1 - \frac{5}{x^2}} = \frac{2 + 0}{1 - 0} = 2$

📋 Summary Table

Function	$\displaystyle \lim_{x \to \infty} f(x)$	$\displaystyle \lim_{x \to -\infty} f(x)$
$f(x) = \frac{1}{x}$	$0$	$0$
$f(x) = \frac{1}{x^2}$	$0$	$0$
$f(x) = e^{-x}$	$0$	$\infty$
$f(x) = \frac{2x^2}{x^2+1}$	$2$	$2$
$f(x) = x^2$	$\infty$	$\infty$

⚠️ Common Mistake

Do not confuse limits at infinity ($x \to \infty$) with infinite limits ($f(x) \to \infty$). They describe different things: one is about where $x$ goes, the other about where $f(x)$ goes.

🔍 Key Takeaways

Limits at infinity describe the end behavior of a function.
For rational functions, compare the degrees of numerator and denominator.
If the degree of the denominator is larger, the limit is $0$.
If degrees are equal, the limit is the ratio of leading coefficients.
If the degree of the numerator is larger, the limit is $\pm\infty$ (no finite horizontal asymptote).

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♾️ Limits at Infinity

📊 Visualizing Limits at Infinity

🧮 How to Compute Limits at Infinity

📋 Summary Table