TITLE: How can I calculate the limit without using the L'Hopital's rule
QUESTION [5 upvotes]: $$ \lim_{x \to 2} \frac{1-\sqrt{1-\sqrt{4x-8}}}{1-\sqrt{1-\sqrt{\frac{x-2}{x+2}}}} $$
I have tried to multiply with conjugates or use auxiliary variables but it did not arrive at all simple

REPLY [2 votes]: Let $u = (4x-8)^{1/2}, v = [(x-2)/(x+2)]^{1/2}.$ The expression equals
$$\frac{u}{v}\left (\frac{1-(1-u)^{1/2}}{u}\right )\big /\left(\frac{1-(1-v)^{1/2}}{v}\right ).$$
Now $u/v = 2(x+2)^{1/2} \to 4$ as $x\to 2^+.$ Because both $u,v \to 0$ as $x\to 2^+,$ both fractions in parentheses approach the same nonzero limit, as you can verify by using the conjugate trick, or simply by recognizing the definition of the derivative of $(1-u)^{1/2}$ at $0.$ It follows that the limit is $4\cdot 1=4.$