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

**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

Brown University

1271 Tutors

California Institute of Technology

2131 Tutors

Carnegie Mellon University

982 Tutors

Columbia University

1256 Tutors

Dartmouth University

2113 Tutors

Emory University

2279 Tutors

Harvard University

599 Tutors

Massachusetts Institute of Technology

2319 Tutors

New York University

1645 Tutors

Notre Dam University

1911 Tutors

Oklahoma University

2122 Tutors

Pennsylvania State University

932 Tutors

Princeton University

1211 Tutors

Stanford University

983 Tutors

University of California

1282 Tutors

Oxford University

123 Tutors

Yale University

2325 Tutors