1.14 Infinite Limits & Vertical Asymptotes

Infinite limits

If f(x) → ±∞ as x → c, then x = c is a vertical asymptote. Writing lim = ∞ describes the behavior, but the limit does not exist as a finite number.

Where they occur

For rational functions, vertical asymptotes occur where the denominator is 0 but the numerator is not. Near the asymptote, one-sided limits may go to +∞ on one side and −∞ on the other — check each side.

Key terms

Infinite limit

f(x) grows without bound as x → c.

Vertical asymptote

The line x = c where f → ±∞.

Where it occurs

Denominator 0, numerator ≠ 0 (rational functions).

One-sided signs

May be +∞ on one side, −∞ on the other.

∞ means DNE

An infinite limit does not exist as a finite number.

Try an AP-style question

lim(x→3⁺) 1/(x − 3) = ?

1. −∞
2. ∞
3. 1/3
4. 0