This discussion focuses on the factoring methods used to solve multivariable limits, specifically the transformation of the term -x into sqrt(x) during the factoring process of the expression sqrt(xy) - x. Participants clarify that -x can be expressed as -x^1, and when factoring out sqrt(x) or x^1/2, the remaining expression simplifies correctly. The conversation emphasizes the importance of recognizing that x and y must be non-negative for these algebraic manipulations to hold true.
PREREQUISITESStudents and educators in mathematics, particularly those focusing on multivariable calculus, algebra enthusiasts, and anyone looking to enhance their understanding of factoring methods in solving limits.
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)?FactChecker said:That is not the same problem. There is a sign difference in the first term of the numerator.
In the work shown, ##\sqrt{xy} - x## was factored into ##\sqrt x(\sqrt y - \sqrt x)##.quittingthecult said: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)?
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.Mark44 said:In the work shown, ##\sqrt{xy} - x## was factored into ##\sqrt x(\sqrt y - \sqrt x)##.
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?Mark44 said:In the work shown, ##\sqrt{xy} - x## was factored into ##\sqrt x(\sqrt y - \sqrt x)##.
Clear?
Why over-complicate things?quittingthecult said: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?
But you don't get separating sqrt(xx) into sqrt(x)*sqrt(x)? Edit: or ## \dfrac {x}{\sqrt x} = \sqrt x ##?quittingthecult said:I get separating sqrt(xy) into sqrt(x)*sqrt(y).
Yes. You could also try multiplying out the factorization @Mark44 showed you and see that you recover what you started with.quittingthecult said: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?