lim x->1 (root x - x^2)/{1 - root x)

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Question from Meghan, a student:

Hi! I have a question from my Calculus textbook that I've been picking at for a while and I'm stuck.
Unfortunately, the answer for this question is not in the back.

lim x->1 (root x - x^2)/{1 - root x).
The problem is I have a square root in the numerator and in the denominator and if I multiply by the conjugate, I can't get rid of both.

All help is greatly appreciated!
-Meghan

We have two responses for you

Hi Meghan,

Multiply the numerator and denominator by the conjugate 1 + √x so that the denominator becomes 1 - x and the numerator becomes √x + x - x² - x²√x = (x - x²) + (√x - x²√x) = x(1 - x) + √x(1 - x)(1 + x). Now simplify the fraction and apply the limit. I got 3 as the limit.

Penny

Meghan,

Try l'Hôpital's Rule.

Stephen La Rocque

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