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

  • #1

Homework Statement



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

[tex]

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.[/tex]

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
 

Answers and Replies

  • #2
phyzguy
Science Advisor
4,483
1,436
Well, can you state what it means for a function to be continuous?
 
  • #3
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
statdad
Homework Helper
1,495
35
Just curious: is the numerator

[tex]
\sqrt{x} -1 + x\sqrt{x-1}
[/tex]

or is it

[tex]
\sqrt{x-1} + x\sqrt{x-1}
[/tex]

It makes a difference in the attack and the answer.
 
  • #5
Hi, thanks for the reply. The numerator is indeed
[tex]

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

[/tex]
 
  • #6
phyzguy
Science Advisor
4,483
1,436
Try using L'Hopital's rule to evaluate the limit of your function as x->1.
 
  • #7
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...
 

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