Find the exact value of c for which f is continuous on its domain

[Charismaztex]

Homework Statement

Let the function f: [1,infinity)-->R

<br /> <br /> f(x)=\left\{ \begin{array}{rcl}<br /> \frac{(\sqrt{x}-1+x\sqrt{x-1}}{\sqrt{x^2-1}} &amp; \mbox{,}<br /> &amp; x&gt;1 \\ <br /> c, x=1<br /> \end{array}\right.

Find the EXACT value of c for which f is continuous on its domain.

Homework Equations

N/A

The Attempt at a Solution

I have sketched the graph on the computer and the first half (x greater than 1) part seems to be a single curve going to infinity on the right. I am not quite sure how to find c.

Thanks in advance,
Charismaztex

 

[phyzguy]

Well, can you state what it means for a function to be continuous?

 

[Charismaztex]

Ah, wait, continuous would mean that c would have to be the limit of the function as it tends towards x=1. So how do we evaluate the limit of this function? Can't seem to find a place to start...

 

[statdad]

Just curious: is the numerator

<br /> \sqrt{x} -1 + x\sqrt{x-1}<br />

or is it

<br /> \sqrt{x-1} + x\sqrt{x-1}<br />

It makes a difference in the attack and the answer.

 

[Charismaztex]

Hi, thanks for the reply. The numerator is indeed
<br /> <br /> \sqrt{x} -1 + x\sqrt{x-1}<br /> <br />

 

[phyzguy]

Try using L'Hopital's rule to evaluate the limit of your function as x->1.

 

[Charismaztex]

Oh yes, that brings up another point I forgot to mention. L'Hopital's rule is banned from this question. The only hint I got was to split the function into its sums...