find the limit as x approaches 1- of [(1-x^2)^1/2] / [(1-x^3)^1/2] aka root(1-x^2)/root(1-x^3)
The Attempt at a Solution
Well I don't really get how to solve this limit using l'hopital's rule if i differentiate both top and bottom i get [1 / 2root(1-x^2)] / [1 / 2root(1-x^3) which is now in the form inf / inf , or if i cross multipl it assumes the same form of my original limit statement with numrator and denominator switched, and if i differentiate top ad bottom again, it goes back to its original form...So basically it seems like one endless loop.
The only wayI can think to get the answer is to remove the square roots (since they apply to both the top and bottom of the fraction), find the limit as x approaches 1 of (1-x^2) / (1-x^3) which is just 2/3. So the original limit must have been root (2/3)
Now that seems to match the answer in the back of the book, but my first instinct was to apply L'hopital's rule even with the root signs, and that didn't work out at all. Can someone explain to me if there is a way to solve this problem by applying L'hopital's rule even with the roots, and, if not, is there other similar situations n which i have to be sneakier in applying L'hopital's rule?