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?
Multivariable limits refer to the concept of finding the limit of a function with multiple variables as the variables approach a specific value. It involves understanding the behavior of a function as it approaches a point in a multi-dimensional space.
Factoring methods allow us to simplify complex expressions and make them easier to evaluate. This is especially important in multivariable limits, where the presence of multiple variables can make the expression difficult to work with. Factoring allows us to break down the expression into simpler terms and ultimately find the limit more easily.
The first step is to factor out any common factors from the expression. Then, we can use algebraic techniques such as factoring by grouping or the difference of squares to further simplify the expression. Next, we can plug in the given values for the variables and evaluate the expression. Finally, we can take the limit as the variables approach the given values to find the final answer.
While factoring methods can be very useful in solving multivariable limits, they may not always work for every type of expression. In some cases, other techniques such as L'Hopital's rule or substitution may be needed to find the limit. It is important to have a good understanding of various techniques and know when to apply them.
Multivariable limits are used in many fields of science and engineering, such as physics, economics, and computer science. They can be used to model and predict the behavior of complex systems, such as the trajectory of a rocket or the efficiency of a financial investment. Understanding multivariable limits can also help in optimizing processes and making informed decisions in various industries.