# limit involving rational function and square root

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.

• Can't you just put $t=\sqrt{\frac{x}{x-1}}$ and solve with $lim_{t\to1}$? – Arpan May 14 '15 at 3:23
• What does the denominator become when you multiply by the conjugate? – parallaxeffect May 14 '15 at 3:24

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.
$$\frac{1-u}{1-\sqrt{u}}=1+\sqrt{u}$$
hint: the answer is $2$ and use $1-A = (1-\sqrt{A})(1+\sqrt{A})$