calculus - Limit $lim_{x to 1} frac{log{x}}{x-1}$ without L'Hôpital

I was wondering if it's possible to identify this limit without using L'Hôpital's Rule:

$$\lim_{x \to 1} \frac{\log{x}}{x-1}$$

Answer

In THIS ANSWER and THIS ONE I showed, without the use of calculus, that the logarithm function satisfies the inequalities

$$\frac{x-1}{x}\le \log(x)\le x-1$$

Therefore, we can write for $x>1$

$$\frac1x \le \frac{\log(x)}{x-1}\le 1$$

and for $x<1$

$$1 \le \frac{\log(x)}{x-1}\le \frac1x$$

whereupon applying the squeeze theorem yields the result $1$.