Where is the mistake in the proof that $\lim_{x\to \infty } \left(\frac{1}{\ln (x)}\right) = 1$?

Question

I think I know where the problem is, I just don't know why it's wrong:

We start with

$$\lim_{x\to \infty }\left(\frac{x}{x\ln (x)}\right).$$

Normally we would just cancel out the $x$'s in the fraction to get

$$\lim_{x\to \infty }\left(\frac{1}{\ln (x)}\right).$$

But, I read somewhere that

$$\lim_{x\to a}\left({f(x)g(x)}\right) = \lim_{x\to a }\left({f(x)}\right)\times\lim_{x\to a }\left({g(x)}\right).$$

Couldn't we then use that to express $\lim_{x\to \infty }\left(\frac{x}{x\ln (x)}\right)$ as

$$\lim_{x\to \infty }\left({\frac{x}{\ln (x)}}\right)\times\lim_{x\to \infty }\left({\frac{1}{x}}\right),$$

and by using L'Hospital's Rule,

$$ \begin{align} \lim_{x\to \infty }\left({\frac{x}{\ln (x)}}\right)\times\lim_{x\to \infty }\left({\frac{1}{x}}\right) &= \lim_{x\to \infty }\left({x}\right)\times\lim_{x\to \infty }\left({\frac{1}{x}}\right)\\ &= \lim_{x\to \infty }\left({\frac{x}{x}}\right)\\ &= \lim_{x\to \infty }\left({1}\right)=1. \end{align} $$

I think the problem occurs when I split the limit into two, but I just can't see why that's a problem.
I would greatly appreciate it if someone could tell me where the problem is along with why it is a problem.

Answer 1

You are right, the problem occurs when you split the limit into two.

---

But I read somewhere that: $\lim_{x \to a} (f(x) g(x)) = \lim_{x \to a} (f(x)) \times \lim_{x \to a} (g(x))$.

This is not true in general. You can split the limit in the above manner only when both the limits on the right-hand side exist.

In this case, you will notice that $\lim_{x \to \infty} \frac{x}{\ln x} = \infty$, so it is not allowed to write

$$ \lim_{x \to \infty} \frac{x}{x \ln x} = \lim_{x \to \infty} \frac{x}{\ln x} \times \lim_{x \to \infty} \frac{1}{x}. $$