Finding a limit without l'hopial

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Homework Statement


lim x→2 ((sqrt(6-x)-2) / (sqrt(3-x) - 1)


Homework Equations


none


The Attempt at a Solution


My daughter's grade 12 mathematics class is not yet using l'hopital, yet this question has been posed as a practice question. I can solve it readily with l'hopital, however without it I'm stumped. When I attempt by multiplying numerator and denominator by (sqrt(3-x) +1 ) predictably I get 0 for my troubles.
 
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Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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