Is this expression infinitesimal, finite, or infinite

[GwtBc]

Homework Statement

Say x is an infinitesimal number on the hyperreal line, is this expression finite, infinite or infinitesimal

Homework Equations

(sqrt(4+x)-2)/x

The Attempt at a Solution

[/B]
My approach so far has been that sqrt(4+x) is (2+y) where y is another infinitesimal and y<x, which seems to lead to the conclusion that this expression is infinite, but the answers say it's finite. Any help?

 

[Samy_A]

Homework Statement

Say x is an infinitesimal number on the hyperreal line, is this expression finite, infinite or infinitesimal

Homework Equations

(sqrt(4+x)-2)/x

The Attempt at a Solution

[/B]
My approach so far has been that sqrt(4+x) is (2+y) where y is another infinitesimal and y<x, which seems to lead to the conclusion that this expression is infinite, but the answers say it's finite. Any help?

If $\sqrt{4+x}=2+y$, where y is infinitesimal, you expression becomes $\frac{y}{x}$. The quotient of two infinitesimals can be anything: infinitesimal, finite but not infinitesimal, or infinite. Your conclusion is therefore premature (and wrong according to your book).

What happens when you muliply both numerator and denominator by $\sqrt{4+x}+2$?

 

[GwtBc]

If $\sqrt{4+x}=2+y$, where y is infinitesimal, you expression becomes $\frac{y}{x}$. The quotient of two infinitesimals can be anything: infinitesimal, finite but not infinitesimal, or infinite. Your conclusion is therefore premature (and wrong according to your book).

What happens when you muliply both numerator and denominator by $\sqrt{4+x}+2$?

$\frac{1}{\sqrt{4+x}+2}$ which is finite. Got it thanks.

You also taught me how to use the maths notation system on the site. Thanks! (You can see I was a noob before)