lim x→1 (1 + x − 2 √ x) / (1 − x) or show that it does not exist. Hint: Put y = √ x. I can't seem to quite figure it out, any help would be appreciated

Question

I've tried factorising the numerator and cancelling out x, multiplying both sides with (1+x) and a few other things, can't quite figure it out. 

Answer 1

Are you familiar with L'Hopital's rule for indeterminate forms?  I suggest using that rule.

 

limit_x→1 (1 + x - 2√x) / (1 - x) = limit_x→1 [1 - 2(1/2)x^-1/2] / (-1)

 

= limit_x→1 [-(1 - 1 / √x)] = -(1 - 1) = 0