# Yet another limit problem!

1. May 2, 2005

### rsnd

I was told to differentiate by definition!!! Function
f = 1/(sqrt(1+x^2));

which gives me the expression
Lim(h->0) {(1/(sqrt(1+(x+h)^2)) - 1/(sqrt(1+x^2)))/h}
problem is...I can’t seem to get rid of the h at the bottom...I’ve tried all teh math packages I have including maple and all of them just seem to complicate things! what should be the simplified version of that function so I don’t get h at the bottom resulting in undefined numbers. or is there a different way to look at this definition problem?

Thanks heaps
K. Civilian

2. May 2, 2005

### matt grime

Why would you want to get rid of it? you need to let it tend to zero....

This ought to help:

Edit:

sqrt(a)-sqrt(b)= (a-b)/(sqrt(a)+sqrt(b))

Last edited: May 2, 2005
3. May 2, 2005

### uart

I suggest you start like this :

( 1 + (x+h)^2 )^(-1/2)

= ( 1 + x^2 + (2x+h)h )^(-1/2)

= (1 + x^2)^(-1/2) ( 1 + (2x+h)/(1+x^2) h )^(-1/2)

Then do a binomial expansion of the "(1 + (2x+h)/(1+x^2) h)^(-1/2)" factor and it's pretty straight forward from there.

4. May 2, 2005

### HallsofIvy

Staff Emeritus
Because if it is in the denominator as it goes to 0 it will cause a lot of trouble!

Provided a-b= 1?

5. May 2, 2005

### matt grime

Noted my mistake.

But the point is that if you use that corrected identity, then you get something where the h on bottom cancels off, and you just have an h on top, let it go to zero and the correct derivative drops out.

Since it's easier to typeset, here's the idea of 1/x

1/(x+h) - 1/x = -h/(x+h)(x)

divide through by h and now let he tend to zero to see that the derivative is -1/x^2

that works here too, but it's a bugger to typeset.

Last edited: May 2, 2005
6. May 2, 2005

### HallsofIvy

Staff Emeritus