limit of $log(x)^{(k+1)}/x$

$$\lim_{x\to \infty} \left(\frac{\log^{k+1}(x)}{x}\right) = \lim_{x\to \infty} \left(\frac{\frac{(k+1)\log^{k}(x)}{x}}{1}\right) = \lim_{x\to \infty} \left(\frac{(k+1)(k)\log^{k-1}(x)}{x}\right) = \lim_{x\to \infty} \left(\frac{(k+1)!}{x}\right) \Rightarrow 0$$

I used L'Hospital to get to

$$\lim_{x\to \infty} \left(\frac{(k+1)(k)\log^{k-1}(x)}{x}\right)$$

But from there I don't understand how to get

$$\lim_{x\to \infty} \left(\frac{(k+1)!}{x}\right)$$

Any help would be appreciated, thanks in advance.