Infinite limits and vertical asymptotes AB & BC

A limit exists when the outputs approach a single real number. Growing without bound is not approaching a number, so an infinite limit is a limit that fails, and the notation records how.

$$\lim_{x\to a^+} f(x) = \infty \quad\text{means}\quad \text{the outputs exceed every bound as } x \text{ nears } a \text{ from the right.}$$

Both readings matter on an exam. Asked "does the limit exist?", the answer is no. Asked "describe the behavior", the answer is $\infty$ with the side specified. Answering "yes, it is $\infty$" conflates the two and is the characteristic error here.

A vertical asymptote at $x = a$ is exactly the geometric statement that at least one one-sided limit there is infinite.

The two sides often run opposite ways, so each needs its own sign analysis:

1. Substitute a value slightly to the left of $a$ and record the sign of the numerator and of the denominator.
2. Divide the signs. Negative over positive is negative, and so on.
3. Repeat slightly to the right.

For $\frac{1}{x-2}$ at $x = 2$: at $x = 1.9$ the denominator is $-0.1$, so the quotient is a large negative number and $\lim_{x\to 2^-} = -\infty$. At $x = 2.1$ it is $+0.1$, so $\lim_{x\to 2^+} = +\infty$.

An even power in the denominator makes both sides agree, since $(x-2)^2$ is positive on both. That is when a single two-sided infinite statement is appropriate.

A vertical asymptote requires the factor to survive after cancellation. Compare at $x = 3$:

1. $\frac{x^2-9}{x-3}$ simplifies to $x+3$, so there is a hole at $(3, 6)$ and no asymptote at all.
2. $\frac{x+1}{x-3}$ cannot be simplified, so there is an asymptote.

Both denominators vanish at 3. Only factoring distinguishes them, which is why "denominator zero, therefore asymptote" is unreliable. The reliable version is: factor, cancel, then read the reduced denominator.

A subtler case: if the numerator's factor has the higher multiplicity, as in $\frac{(x-3)^2}{x-3}$, the factor cancels completely and a hole remains. If the denominator's is higher, as in $\frac{x-3}{(x-3)^2}$, some of it survives and an asymptote remains.

Three conventions save marks. State the side whenever the two differ, because a bare $\infty$ hides a sign disagreement. Say the limit does not exist when asked about existence, even while describing it as infinite. And do not write $\pm\infty$ as a value: it is not one, and where the sides differ the two-sided limit simply fails.

The connection to the graph is worth stating plainly. Each one-sided infinite limit is one branch of the curve running along the asymptote, so a full description names two behaviors, not one.