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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.

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

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  • $\begingroup$ Thanks. Understand it now! $\endgroup$ – user1.0 May 14 '15 at 3:49
  • $\begingroup$ You are welcome! $\endgroup$ – homegrown May 14 '15 at 3:50
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Hint:

$$\frac{1-u}{1-\sqrt{u}}=1+\sqrt{u}$$

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

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