# Conjugate the limit

#### walker242

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

1. The problem statement, all variables and given/known data
Calculate the limit of $$\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$$.

2. Relevant equations
As above.

3. The attempt at a solution
Have tried to multiplicate with the conjugate.

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#### cristo

Staff Emeritus
Re: $\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$

Have tried to multiplicate with the conjugate.
Ok, what did you get? Note that you must show your work in order to get help here.

#### walker242

Re: $\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$

$$\lim_{x\to 1} \frac{\sqrt{x}-1}{x-1} = \lim_{x\to 1} \frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1)}{\left(x-1\right)\left(\sqrt{x}+1\right)} = \lim_{x\to 1} \frac{x-1}{x\sqrt{x}+x-\sqrt{x}-1} = \lim_{x\to1}\frac{x-1}{\sqrt{x}\left(x-1\right)+x-1} = \lim_{x\to1}\frac{x-1}{(x-1)(\sqrt{x}+1)} = \frac{1}{2}$$

So in essence, disregard me, for I am retarded. :P

#### HallsofIvy

Re: $\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$

For a retarded person, remarkably good at limits!

#### rrogers

Re: $\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$

#### HallsofIvy

Re: $\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$

Why? That's like using a sledgehammer to crack a walnut. Walker242's solutions is excellent- especially because it is his solution!

#### rrogers

Re: $\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$

Why?
His solution is very good.
So I have no overriding reason; but LHopital is more generic.
But I am into generic, versus tricky.

"Conjugate the limit"

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