# Homework Help: Find the exact value of c for which f is continuous on its domain

1. Mar 14, 2010

### Charismaztex

1. The problem statement, all variables and given/known data

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

$$f(x)=\left\{ \begin{array}{rcl} \frac{(\sqrt{x}-1+x\sqrt{x-1}}{\sqrt{x^2-1}} & \mbox{,} & x>1 \\ c, x=1 \end{array}\right.$$

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

2. Relevant equations

N/A

3. 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.

Charismaztex

2. Mar 14, 2010

### phyzguy

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

3. Mar 15, 2010

### 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...

4. Mar 15, 2010

Just curious: is the numerator

$$\sqrt{x} -1 + x\sqrt{x-1}$$

or is it

$$\sqrt{x-1} + x\sqrt{x-1}$$

It makes a difference in the attack and the answer.

5. Mar 15, 2010

### Charismaztex

Hi, thanks for the reply. The numerator is indeed
$$\sqrt{x} -1 + x\sqrt{x-1}$$

6. Mar 15, 2010

### phyzguy

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

7. Mar 15, 2010

### 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...