Understanding Multivariable Limits: Solving with Factoring Methods

[guyvsdcsniper]

Homework Statement

Find the limit

Relevant Equations

lim(x,y)->(0,0)

I do not understand how they got the -x in the numerator to turn into a sqrt(x) when factoring to solve this multivariable function. Could some help me understand?

 

[FactChecker]

That is not the same problem. There is a sign difference in the first term of the numerator.

 

[guyvsdcsniper]

That is not the same problem. There is a sign difference in the first term of the numerator.

I missed that. I need to edit my post. The 1st problem in the 2nd image is what I am questioning. How did they get -x to become sqrt(x)?

 

[Mark44]

I missed that. I need to edit my post. The 1st problem in the 2nd image is what I am questioning. How did they get -x to become sqrt(x)?

In the work shown, $\sqrt{xy} - x$ was factored into $\sqrt x(\sqrt y - \sqrt x)$.

Clear?

 

[guyvsdcsniper]

In the work shown, $\sqrt{xy} - x$ was factored into $\sqrt x(\sqrt y - \sqrt x)$.

Yea that is what's getting me. I get separating sqrt(xy) into sqrt(x)*sqrt(y). I just don't see how the -x turns into sqrt(x) when factoring.

 

[guyvsdcsniper]

In the work shown, $\sqrt{xy} - x$ was factored into $\sqrt x(\sqrt y - \sqrt x)$.

Clear?

Oh wait. The -x is technically -x^1 and sqrt of x is technically x^1/2. so if pull a x^1/2 you are left with a x^1/2. Is that correct?

 

[pbuk]

Oh wait. The -x is technically -x^1 and sqrt of x is technically x^1/2. so if pull a x^1/2 you are left with a x^1/2. Is that correct?

Why over-complicate things?

I get separating sqrt(xy) into sqrt(x)*sqrt(y).

But you don't get separating sqrt(xx) into sqrt(x)*sqrt(x)? Edit: or $\dfrac {x}{\sqrt x} = \sqrt x$?

Note that we must be sure that x (and y) are non-negative for these manipulations.

 

[vela]

Oh wait. The -x is technically -x^1 and sqrt of x is technically x^1/2. so if pull a x^1/2 you are left with a x^1/2. Is that correct?

Yes. You could also try multiplying out the factorization @Mark44 showed you and see that you recover what you started with.