Infinite Limits

Infinite Limits: When Functions Grow Without Bound

An infinite limit occurs when $f(x)$ becomes arbitrarily large (positive or negative) as $x$ approaches some value $a$. We write $\lim_{x\to a} f(x) = +\infty$ or $\lim_{x\to a} f(x) = -\infty$ as shorthand notation — this does not mean the limit exists in the traditional sense, but rather that the function is unbounded near $a$. This behavior reveals the presence of a vertical asymptote.

Key Definitions

Why This Happens: The Denominator Approaches Zero

Infinite limits most commonly arise with rational functions. When a denominator approaches $0$ but the numerator approaches a nonzero value, division by a tiny number produces huge values. Consider $f(x) = \dfrac{1}{x}$:

• As $x \to 0^+$: $x$ is a small positive number, so $\dfrac{1}{x} \to +\infty$
• As $x \to 0^-$: $x$ is a small negative number, so $\dfrac{1}{x} \to -\infty$

Since the one-sided limits differ, the two-sided limit doesn't exist, but $x=0$ is a vertical asymptote.

Finding the Sign: A Strategy

For $\lim_{x\to a} \dfrac{p(x)}{q(x)}$ where $q(a) = 0$:

1. Check if $p(a) \neq 0$ — if true, expect an infinite limit
2. Test the sign of the denominator on each side of $a$: is it small positive or small negative?
3. Combine signs: $\dfrac{\text{nonzero}}{0^+} = +\infty$; $\dfrac{\text{nonzero}}{0^-} = -\infty$

⚠️ Critical Pitfall: If both numerator and denominator vanish at $a$, factor and cancel first — you may have a removable hole, not an asymptote. Always simplify before concluding an infinite limit exists.