Problems with multiplying by the conjugate of both the numerator and denominator

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Question description

This is a Calculus 1 problem, but I am having problems with the algebra portion. So far, I have determined that I need to multiply this problem by the conjugate of BOTH the numerator and denominator, but I've never done that. Once I get the fraction all worked out and cleaned up, I know how to handle the limit portion.

 Evaluate the limit of:

(lim x approaches 2)(((6-x)^(1/2))-2) / (((3-x)^(1/2))-1)  

 Here is my work:

 (((6-x)^(1/2))-2)/(((3-x)^(1/2))-1) * (((6-x)^(1/2))-2)/(((6-x)^(1/2))-2)

 => (((6-x)^2)-((4)^2))/(((6-x)^(1/2))-2)/(((3-x)^(1/2))-1)

 Then do I multiply out the top, and then multiply by the conjugate of the numerator?

=> ((-x^2)-12x+36)/(((6-x)^(1/2))-2)/(((3-x)^(1/2))-1) * (((3-x)^(1/2))-1)/(((3-x)^(1/2))-1)

This is where I'm really confused. How do I multiply a polynomial by the conjugate? Would the numerator be ((-x^2)(3-(x^(1/2)))-12x((3-(x^(1/2))+36((3-x)^(1/2))-x^2+12x-36?

Now what? I appreciate any and all help. Please let me know where I went wrong, and what I need to do to work out the problem.

We are not using L'Hôpital's Rule yet, so that doesn't help.

Tutor Answer

(Top Tutor) Daniel C.
School: University of Virginia
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