Topic 1.14 Connecting Infinite Limits and Vertical Asymptotes: use infinite limits to identify vertical asymptotes and describe behavior near them with correct sign analysis.

What this topic is asking

The College Board (Topic 1.14) wants you to connect an infinite limit to a vertical asymptote. When a function's outputs grow without bound near a point, the line $x = a$ is a vertical asymptote, and you must use sign analysis to say whether the function heads to $+\infty$ or $-\infty$ on each side.

Infinite limits

This happens for a rational function where the denominator approaches $0$ but the numerator approaches a nonzero number. (If both approach $0$, you have a $\frac{0}{0}$ form to resolve first - that may be a hole instead.)

Sign analysis for the direction

Where asymptotes come from

A vertical asymptote of a rational function sits at each zero of the denominator that does not cancel with the numerator. If a factor cancels, that point is a removable hole, not an asymptote (Topic 1.10). So always factor first, cancel removable factors, and the remaining denominator zeros are the asymptotes.

Holes versus asymptotes: factor first

The single most important habit before declaring a vertical asymptote is to factor and cancel first. A zero of the denominator only produces an asymptote if it survives cancellation; if the same factor appears in the numerator and cancels, that point is a removable hole instead, where the limit is finite. For example, $\frac{x - 2}{x^2 - 4} = \frac{x - 2}{(x-2)(x+2)} = \frac{1}{x+2}$ has a hole at $x = 2$ (limit $\frac14$) but a genuine asymptote at $x = -2$. Skipping the factoring step leads to the common error of calling every denominator zero an asymptote. After cancelling, the surviving denominator zeros are the asymptotes, and at each you do sign analysis. This connects the topic tightly to the classification of discontinuities and to evaluating $\frac{0}{0}$ forms by algebraic manipulation.

Reporting the result

When a limit is infinite, state the direction with the $\pm\infty$ symbol and, if asked, note that the (finite) limit does not exist. On a graph, an infinite limit shows the curve hugging the vertical line $x = a$, shooting up or down. Pairing the one-sided behaviors gives the full picture near the asymptote. A useful shortcut for the direction: an odd-power factor in the denominator (like a single $(x - a)$) changes sign across $a$, so the two sides give opposite infinities, whereas an even-power factor (like $(x - a)^2$) keeps the same sign on both sides, so both one-sided limits share the same infinity. Recognizing the parity of the factor lets you predict the behavior before doing the full sign table.