Problems when solving an infinity limit: (1)/(√(x^2-x)-x)

Sirus Glaceon

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Dec 29, 2016
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When we are trying to solve limits involving different powers on numerator and denominator, we will just simplily divide the whole function with the highest power. So I tried to apply this method in a limit as the following
lim x-∞ (1)/(√(x^2-x)-x), diving the both the denominator and numerator with √(x^2). However the result is wrong. I know that this limit has to be solved by multilying something like (a-b)(a+b). But why can't we divide a variable? And under what situation can we divide a variable in a infinity limit?
Many thanks and happy new year!
 
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There is absolutely no reason whatsoever to use L'Hospital's rule. Here is you original question:
limx1x2xx=limxx2x+xx=2\displaystyle \displaystyle{{\lim _{x \to \infty }}\frac{1}{{\sqrt {{x^2} - x} - x}} = {\lim _{x \to \infty }}\frac{{\sqrt {{x^2} - x} + x}}{{ - x}} = - 2} SEE HERE

Both the original post and the reply are hopelessly obscure notation wise. Such messes are almost impossible to read. For example, lim(x-infinity)?? How does that mean x approaches infinity?
Look at my reply. Using simple LaTeX coding it is completely readable.

As for the other reply, not only is it hard to read it is also miss-leading. Can you see how a simple algebra problem is turned it a complicated mess?
It is simple algebra because multiplying by x2x+xx2x+x\displaystyle \frac{{\sqrt {{x^2} - x} + x}}{{\sqrt {{x^2} - x} + x}} is all it takes.
 
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