Limit of (1/(x√(x+1)) - 1/x) as x approaches 0

[Manarius]

This is just a problem I came across while reviewing basic calculus.

Homework Statement

Find the limit as x approaches 0 of f(x)=(1/(x(x+1)^1/2)) - (1/x)

Homework Equations

The Attempt at a Solution

My problem here is really more of an algebra problem than a calculus problem. I cannot for the life of me remember how to get the limit of the denominator of either term to not equal 0. I've tried everything I can think of (rationalizing the denominator, etc.), to no avail.

I could find no tutorials either online or in my book that use an example quite like this one. It's truly maddening, especially because I know I should have learned this 6 years ago in Algebra.

Thanks in advance. Hope you can help.

 

[ehild]

Do you mean

$$f(x)=\frac{1}{x\sqrt{x+1}}-\frac{1}{x}$$

?

If yes, factor out 1/x and multiply and divide f(x) by [1/sqrt(x+1)+1].

ehild

 

[Manarius]

I think I must be misunderstanding what you're saying. Either that or I did the math wrong multiple times.

Would you mind clarifying what exactly you mean? Thanks.

 

[ehild]

Use the notation 1/√(x+1)=a.

f(x)=(1/x) (a-1)

Multiply and divide by (a+1)

f(x)=(1/x)[(a-1)(a+1)]/(a+1)

f(x)=(1/x)(a²-1)/(a+1).

Replace back 1/√(x+1) for a and simplify. Find the limit.