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

• Charismaztex
In summary, the function f(x) is given as a piecewise function and the task is to find the exact value of c for which the function is continuous on its domain. The numerator of the function is \sqrt{x} -1 + x\sqrt{x-1}. The approach of using L'Hopital's rule is not allowed and the only hint given is to split the function into its sums.

## Homework Statement

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.

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.

Charismaztex

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

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...

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.

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

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

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...

## What is the definition of continuity for a function?

The definition of continuity for a function f at a point c is that the limit of f(x) as x approaches c is equal to the value of f at c. This means that the function has a smooth and unbroken graph at that point, without any abrupt changes or jumps.

## Why is it important to find the exact value of c for which a function is continuous?

It is important to find the exact value of c for which a function is continuous because this value can affect the behavior and properties of the function. It can also help in determining the intervals on which the function is continuous, and in turn, understanding the overall behavior of the function.

## How can I determine the value of c for which a function is continuous?

To determine the value of c for which a function is continuous, you can use the definition of continuity and solve for c. This typically involves finding the limit of the function at the point c and equating it to the value of the function at c. You can also use graphical methods or algebraic techniques to find the value of c.

## What happens if there is no value of c for which a function is continuous?

If there is no value of c for which a function is continuous, then the function is considered to be discontinuous at all points on its domain. This means that the function has abrupt changes or jumps at every point, and it is not possible to find a value of c that satisfies the definition of continuity.

## How can I use the value of c to determine the continuity of a function over a given interval?

If you have found the value of c for which a function is continuous, you can use it to determine the intervals on which the function is continuous. This can be done by checking if the limit of the function at the endpoints of the interval is equal to the value of the function at those points. If this is true, then the function is continuous over that interval. If not, then the function is discontinuous over that interval.