<s>[INST] Simple limit problem with squares I'm doing a refreshment course in math but I'm stuck with some problem. Although this problem doesn't look hard I don't know what I'm doing wrong. 
$$\lim_{x \to 4} \frac{\sqrt{x-3}-1}{2\sqrt{2}-\sqrt{x^2-3x+4}}$$ 
I have tried to take the conjuct of both squares but I still got the indeterminate form of $\frac{0}{0}$.
Thanks in advance... 
 [/INST] Multiply numerator and denominator by $\displaystyle (2\sqrt{2}+\sqrt{x^2-3x+4})(\sqrt{x-3}+1)$:
$$\lim_{x \to 4} \frac{\sqrt{x-3}-1}{2\sqrt{2}-\sqrt{x^2-3x+4}}=\lim_{x \to 4} \frac{ (2\sqrt{2}+\sqrt{x^2-3x+4})(\sqrt{x-3}+1)(\sqrt{x-3}-1)}{(2\sqrt{2}-\sqrt{x^2-3x+4}) (2\sqrt{2}+\sqrt{x^2-3x+4})(\sqrt{x-3}+1)}= \\ = \lim_{x \to 4}\frac{(2\sqrt{2}+\sqrt{x^2-3x+4})(x-3-1)}{-(x^2-3x-4)(\sqrt{x-3}-1)}=\lim_{x \to 4}\frac{(x-4)(2\sqrt{2}+\sqrt{x^2-3x+4})}{-(x+1)(x-4)(\sqrt{x-3}+1)}=\\=\lim_{x \to 4}\frac{(2\sqrt{2}+\sqrt{x^2-3x+4})}{-(x+1)(\sqrt{x-3}+1)}=\frac{-(2\sqrt{2}+2\sqrt{2})}{5(\sqrt{2}+1)}$$
</s>