Limit involving square roots

Hi there,

I am trying to see why this function goes to zero at the limit of large n (for constant a):

With some factoring out, I've arrived at this:

Where the first (root) term in the bracket should go to 1, and the last to 0, thus the bracket should approach zero in the limit. However, for some reason I don't feel quite comfortable with my method and reasoning. Don't I have to worry about the rate the two terms of the product approach their respective limits? Is there a better way to express the function to make this nuance clear?

Thanks,

Chris

Re: Limit involving square roots

Quote:

Originally Posted by

**entropyslave** Hi there,

I am trying to see why this function goes to zero at the limit of large n (for constant a):

With some factoring out, I've arrived at this:

Where the first (root) term in the bracket should go to 1, and the last to 0, thus the bracket should approach zero in the limit. However, for some reason I don't feel quite comfortable with my method and reasoning. Don't I have to worry about the rate the two terms of the product approach their respective limits? Is there a better way to express the function to make this nuance clear?

You should not feel comfortable! Yes, the bracket approaches 0 but then you are multiplying it by a term that goes to infinity- you have an "indeterminate" of the form .

Instead do what you typically do with problems involving square roots- multiply both numerator and denominator by the "conjugate":

Re: Limit involving square roots

Quote:

Originally Posted by

**HallsofIvy** You should not feel comfortable! Yes, the bracket approaches 0 but then you are multiplying it by a term that goes to infinity- you have an "indeterminate" of the form

.

Instead do what you typically do with problems involving square roots- multiply both numerator and denominator by the "conjugate":

I see, thanks for making this clearer to me!

So there was a typo in my original formula - it should of read (with the 2 as part of the first term):

So now following your method, I get:

And now this already clearly goes to zero, but I think it is even clearer if I apply a similar factorisation as my original post:

Which for large n looks like

Thanks for your help!