SOLUTION: Hello I am completely stuck on this question. I would appreciate any help possible! Also, I can't use L'Hopital's rule on this question. lim x->2, ((sqrtx-sqrt2)/(x-2)) Th

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson  -> Lesson -> SOLUTION: Hello I am completely stuck on this question. I would appreciate any help possible! Also, I can't use L'Hopital's rule on this question. lim x->2, ((sqrtx-sqrt2)/(x-2)) Th      Log On


   

Question 944774: Hello
I am completely stuck on this question. I would appreciate any help possible! Also, I can't use L'Hopital's rule on this question.
lim x->2, ((sqrtx-sqrt2)/(x-2))
Thanks!

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Recognize that the denominator is the difference of two squares.
Use a substitution,sqrt%28x%29=u%29 and sqrt%282%29=a
So then,
%28u-a%29%2F%28u%5E2-a%5E2%29=%28u-a%29%2F%28%28u-a%29%28u%2Ba%29%29
%28u-a%29%2F%28u%5E2-a%5E2%29=1%2F%28u%2Ba%29
So,
%28sqrt%28x%29-sqrt%282%29%29%2F%28x-a%29=1%2F%28sqrt%28x%29%2Bsqrt%282%29%29
and then the limit,

lim%28x-%3E2%2C%28sqrt%28x%29-sqrt%282%29%29%2F%28x-2%29%29=%281%2F2%29%281%2Fsqrt%282%29%29
lim%28x-%3E2%2C%28sqrt%28x%29-sqrt%282%29%29%2F%28x-2%29%29=%281%2F2%29%28sqrt%282%29%2F2%29
lim%28x-%3E2%2C%28sqrt%28x%29-sqrt%282%29%29%2F%28x-2%29%29=highlight%28sqrt%282%29%2F4%29