1.14 Connecting Infinite Limits and Vertical Asymptotes

Infinite limits describe unbounded behavior

When a function's output grows without bound as x approaches a value c, we write lim(x→c) f(x) = ∞ or -∞. This means the limit does not exist as a finite number — the function is increasing or decreasing without bound near c, often due to division by a value approaching zero.

Vertical asymptotes come from infinite limits

A vertical asymptote occurs at x = c if lim(x→c⁻) f(x) or lim(x→c⁺) f(x) is ∞ or -∞. For rational functions, this typically happens where the denominator equals zero but the numerator does not (no common factor cancels). The graph approaches but never touches the vertical line x = c.

Checking one-sided behavior matters

The sign of an infinite limit can differ from the left and right sides of x = c, causing the function to shoot up on one side and down on the other. Always test the sign of the function just left and right of the suspected asymptote to describe the behavior accurately.

Key terms

Infinite limit

lim(x→c) f(x) = ∞ or -∞ means f(x) grows or decreases without bound as x approaches c.

Vertical asymptote

A vertical line x = c that the graph approaches but never touches, occurring when a one-sided limit at c is infinite.

Condition for vertical asymptote (rational function)

Denominator equals zero at x = a while the numerator is nonzero (no cancellation).

Removable discontinuity vs. vertical asymptote

A hole occurs when a common factor cancels; a vertical asymptote occurs when it doesn't and the limit is infinite.

How to confirm a vertical asymptote

Check that at least one one-sided limit (left or right) approaches +∞ or -∞ as x approaches the value.

Try an AP-style question

lim(x→3⁺) 1/(x − 3) = ?

1. 1/3
2. 0
3. ∞
4. −∞