Limit of ln x as x Approaches 0

Reading the answer off the exponential

$\ln x$ is the inverse of $e^{x}$, so asking what the logarithm does near 0 is asking which exponents make $e^{y}$ tiny. The statement $\ln x = y$ is the same statement as $e^{y} = x$.

To force $e^{y}$ below $0.001$ you need $y$ under about $-6.9$; to force it below $10^{-100}$ you need $y$ under about $-230$. There is no floor. For any bound you name, some positive $x$ sends $\ln x$ beneath it, so the values are unbounded below.

Watch the pace in that table. Cutting $x$ by a factor of 10 lowers $\ln x$ by only about $2.303$. The descent never stops, but it is slow, and that slowness is why $x \ln x$ still goes to 0.

Why substitution fails, and why nothing here is indeterminate

Substitution gives $\ln 0$, which is not a number and not a form. Expressions such as $\frac{0}{0}$ and $0^0$ are indeterminate because two competing behaviours are in play. Here nothing competes: the input is outside the domain, and the function is unbounded as it approaches the edge of that domain.

So there is no algebra to do and no rule to reach for. The value comes from end behaviour, the same way $\lim_{x \to 0^+} \frac{1}{x} = \infty$ does. This is Topic 1.14, connecting infinite limits and vertical asymptotes, which puts it in Unit 1 (10 to 15 percent of the AB exam, 5 to 10 percent of BC).

An infinite limit means the limit does not exist as a real number. Writing $-\infty$ still says more than writing that it does not exist, because it names the direction, and AP rubrics expect that notation.

The mistake students make

• Writing $\lim_{x \to 0} \ln x$ without the superscript. The two-sided limit is not merely infinite, it is undefined as a question, since there are no function values to the left of 0.
• Answering 0, from a half-memory that the logarithm has a zero. It does, at $x = 1$, not at $x = 0$.
• Mixing this up with the behaviour at the far end. $\ln x \to \infty$ as $x \to \infty$ and $\ln x \to -\infty$ as $x \to 0^+$, so both ends run away, in opposite directions.
• Calling $x = 0$ a removable discontinuity. It is an infinite discontinuity, and no value assigned at 0 would patch $\ln x$ into a continuous function.