AP Calculus AB and BC

Limit of x / ln x at Infinity Is Infinity

Q: What about ln x / x?

That tends to 0: the same fact stated as a reciprocal.

The limit of x over ln x as x approaches infinity is infinity. One pass of L'Hopital's rule turns it into x, which grows without bound. A logarithm grows more slowly than any positive power of x, so the ratio is unbounded.

\lim_{x \to \infty} \frac{x}{\ln x} = \infty

Settled by L'Hopital's rule.

One pass

\lim_{x \to \infty}\frac{x}{\ln x} \;\overset{\text{H}}{=}\; \lim_{x \to \infty}\frac{1}{1/x} = \lim_{x \to \infty}x = \infty

The reciprocal limit $\frac{\ln x}{x} \to 0$ is the same fact stated the other way, and it is the one that appears more often.

How slowly logarithms grow

At $x = 10^{6}$ the ratio is already about $72{,}000$ , and it keeps climbing. That gap is why $\ln x$ sits at the bottom of the growth ordering, below even $\sqrt{x}$ .

This limit is also the shape of the prime counting function's density, which is where it shows up outside calculus.

The mistakes students make

Answering $1$ by treating both parts as comparable.
Answering $0$ by confusing it with $\frac{\ln x}{x}$ , the reciprocal.

Not sure which technique a limit wants?

The Limit Method Chooser walks the decision from direct substitution through factoring, the conjugate, and L'Hopital, and says why each one applies or fails.

Frequently asked questions

What is the limit of x / ln x at infinity?

It is $\infty$ .

What about ln x / x?

That tends to $0$ : the same fact stated as a reciprocal.