Infinite Limits – AP Calculus

Sometimes as $x$ approaches a finite number $a$, the function grows without bound — it gets larger and larger in the positive or negative direction. This is called an infinite limit, and it tells us that the graph has a vertical asymptote at $x = a$.

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

📌 Vertical Asymptotes

If $\displaystyle \lim_{x \to a} f(x) = \pm\infty$, then the line $x = a$ is a vertical asymptote of the graph of $f$.

📊 Visualizing Infinite Limits

Use the interactive graph below. Notice how the function shoots up or down as $x$ approaches the asymptote:

🧮 Common Patterns for Infinite Limits

The most common place infinite limits appear is with rational functions where the denominator approaches zero but the numerator does not.

📐 Rule of Thumb

For $f(x) = \frac{p(x)}{q(x)}$ where $p(a) \neq 0$ and $q(a) = 0$:

The limit will be $\pm\infty$ (infinite)
The sign depends on the sign of $p(a)$ and the behavior of $q(x)$ as $x \to a$ from left and right

Example 1Basic Reciprocal: $\frac{1}{x}$

Find $\displaystyle \lim_{x \to 0^+} \frac{1}{x}$ and $\displaystyle \lim_{x \to 0^-} \frac{1}{x}$.

As $x \to 0^+$ (positive small numbers), $\frac{1}{x}$ is positive and gets huge → $\infty$
As $x \to 0^-$ (negative small numbers), $\frac{1}{x}$ is negative and gets huge in magnitude → $-\infty$
Since the one-sided limits differ, $\displaystyle \lim_{x \to 0} \frac{1}{x}$ does not exist (DNE)

Example 2Even Power: $\frac{1}{x^2}$

Find $\displaystyle \lim_{x \to 0} \frac{1}{x^2}$.

As $x \to 0$ from either side, $x^2 > 0$, so $\frac{1}{x^2}$ is always positive and grows without bound
$\displaystyle \lim_{x \to 0} \frac{1}{x^2} = \infty$ (both one-sided limits agree)

Example 3Shifted Asymptote

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

As $x \to 2^-$ (values slightly less than 2), $x-2$ is negative and small → $\frac{1}{x-2} \to -\infty$
As $x \to 2^+$ (values slightly greater than 2), $x-2$ is positive and small → $\frac{1}{x-2} \to \infty$
Vertical asymptote at $x = 2$

Example 4Rational Function with Polynomial Numerator

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

As $x \to 1$, numerator $x+2 \to 3 \neq 0$
Denominator $(x-1)^2 \to 0^+$ (always positive because of square)
Positive numerator divided by a positive small number → $\infty$
$\displaystyle \lim_{x \to 1} \frac{x+2}{(x-1)^2} = \infty$

Example 5Determining the Sign

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

Check sign from left ($x \to -2^-$): $x-1$ is negative, $x+2$ is negative → negative/negative = positive → $\infty$
Check sign from right ($x \to -2^+$): $x-1$ is negative, $x+2$ is positive → negative/positive = negative → $-\infty$
So $\displaystyle \lim_{x \to -2^-} = \infty$, $\displaystyle \lim_{x \to -2^+} = -\infty$, two-sided limit DNE

📋 Quick Reference: Sign Analysis

Denominator behavior	Numerator sign	Result
$q(x) \to 0^+$	$p(a) > 0$	$\infty$
$q(x) \to 0^+$	$p(a) < 0$	$-\infty$
$q(x) \to 0^-$	$p(a) > 0$	$-\infty$
$q(x) \to 0^-$	$p(a) < 0$	$\infty$

⚠️ Important Distinction

Do not confuse infinite limits ($f(x) \to \infty$) with limits at infinity ($x \to \infty$). They are different concepts!

Infinite limits: $x$ approaches a finite number, $f(x)$ blows up
Limits at infinity: $x$ blows up, $f(x)$ approaches a finite number (or also blows up)

🔍 Key Takeaways

Infinite limits occur when a function grows without bound as $x \to a$.
They indicate the presence of a vertical asymptote at $x = a$.
For rational functions: if denominator → 0 and numerator ≠ 0, the limit is infinite.
Always check the sign (from left and right) to determine $\infty$ or $-\infty$.
One-sided infinite limits can differ, in which case the two-sided limit does not exist.

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📈 Infinite Limits

📊 Visualizing Infinite Limits

🧮 Common Patterns for Infinite Limits

📋 Quick Reference: Sign Analysis