[Math] limit involving rational function and square root

Question

When working some exercise problems in my calc book, I came across this limit in which I do not know how to tackle. It is
$$\lim_{x\to\infty}\frac{1-\frac{x}{x-1}}{1-\sqrt{\frac{x}{x-1}}}$$
I feel like there is a trick to this one, maybe use L'Hopital's rule or something. I tried to multiplying by the conjugate but it turned ugly real fast. Any tips will be helpful.

Answer 1

Hint

Let $t=\sqrt{\frac{x}{x-1}}.$ Now as $x\to\infty,$ we have $t\to 1$. Now just factor and limit is much easier.