Limit of $\frac{\ln\left(x\right)}{x}$ as $x \to \infty$

Q: What is the limit of ln(x) / x as x approaches ∞?

The limit of (ln ≤ft(x))/(x) as x → is 0.

The limit of $\frac{\ln\left(x\right)}{x}$ as $x \to \infty$ is $0$ .

\lim_{x \to \infty} \frac{\ln\left(x\right)}{x} = 0

Explain it step by step in chat Make a short video of it

Graph of ln(x) / x near x = ∞. The curve levels off at 0 as x grows.

How to work it out

1
We want what ln(x) / x approaches as x → ∞.
$\lim_{x \to \infty} \frac{\ln\left(x\right)}{x}$
2
Direct substitution gives an indeterminate form: ∞/∞. It does not tell us the limit.
3
L'Hôpital's rule: when the form is ∞/∞, the limit equals the limit of the ratio of the derivatives.
$\lim \frac{\ln\left(x\right)}{x} = \lim \frac{\frac{1}{x}}{1}$
4
Method: L'Hôpital: (1/x)/1 → 0; logarithms grow slower than any power.
5
Result:
$\lim_{x \to \infty} \frac{\ln\left(x\right)}{x} = 0$

LaTeX source of the formulas on this page

\[ \lim_{x \to \infty} \frac{\ln\left(x\right)}{x} = 0 \]
\[ \lim_{x \to \infty} \frac{\ln\left(x\right)}{x} \]
\[ \lim \frac{\ln\left(x\right)}{x} = \lim \frac{\frac{1}{x}}{1} \]

Values of ln(x) / x as x → ∞ (computed)

x	f(x)	Distance from the limit
10	0.23025851	2.30e-1
100	0.046051702	4.61e-2
1000	0.0069077553	6.91e-3
10000	0.00092103404	9.21e-4
100000	0.00011512925	1.15e-4
1000000	0.000013815511	1.38e-5

Questions

What is the limit of ln(x) / x as x approaches ∞?

The limit of $\frac{\ln\left(x\right)}{x}$ as $x \to \infty$ is $0$ .

Which form is it?

∞/∞.

Limit of ln⁡(x)x\frac{\ln\left(x\right)}{x}xln(x)​ as x→∞x \to \inftyx→∞

How to work it out

Values of ln(x) / x as x → ∞ (computed)

Questions

More well-known limits

Limit of $\frac{\ln\left(x\right)}{x}$ as $x \to \infty$