Infinite Limits

An infinite limit is simply what we call it when a function shoots off the top (or bottom) of the graph as you get near a certain $x$. The name is a little misleading: the limit doesn't actually exist, and "infinite" just describes the way it fails. It's good for spotting where a formula breaks down, like a lens that can't focus or a cost that can't be paid. Formulas like $\lim_{x\to0^+}\frac1x = \infty$ just mean $x\to0^+$ (as $x$ shrinks toward $0$ from the right), $\frac1x$ (one divided by that tiny number) and $= \infty$ (gets bigger than any number you name). Dividing by smaller and smaller numbers gives bigger and bigger answers, and that's really all there is to it.

Back when we listed the ways a limit can fail to exist, one of them was a blow-up, like $\frac{1}{x^2}$ near 0. Blow-ups are common enough that they get their own notation. If the values grow past every bound as $x \to a$, we write $\lim_{x\to a} f(x) = \infty$ (or $-\infty$ if they head down).

A common mistake. It's tempting to read "$= \infty$" as "the limit exists and equals infinity." It doesn't. $\infty$ isn't a number, and the outputs aren't settling on anything. Writing $= \infty$ is a more informative way of saying the limit doesn't exist: it tells you how it fails.

Sign analysis. Last lesson, $\frac00$ meant "simplify and look again." A different situation is when the top heads to a nonzero number and only the bottom heads to $0$. Think about what dividing by a tiny number does: $\frac{1}{0.001} = 1000$, $\frac{1}{0.000001} = 1{,}000{,}000$. The quotient blows up. All that's left is to figure out the sign, and that comes from the signs of the pieces:

• $\dfrac{\text{positive}}{\to 0^+} \to +\infty$, $\quad\dfrac{\text{positive}}{\to 0^-} \to -\infty$, and so on.

This is where one-sided limits pay off: the bottom often approaches $0$ through positive numbers on one side and negative numbers on the other.

Vertical asymptotes sit wherever a one-sided limit is $\pm\infty$. For rational functions, look for zeros of the denominator that don't cancel with the numerator (if they do cancel, you just get a hole, like before). Other ones you'll see a lot: $\ln x$ at $x = 0^+$ (it goes to $-\infty$) and $\tan x$ at $x = \frac\pi2$.

Coulomb's law blows up near a point charge, a resonance amplitude blows up as damping vanishes, and the cost of removing the last bit of a pollutant often grows like $\frac{1}{1 - p}$ as $p \to 1$.