Note that $$\lim_{x\to2}\frac{\sqrt{1+\sqrt{x+2}-\sqrt3}}{x-2}$$ The deonominator goes to zero, but the numerator goes to $\sqrt{3-\sqrt{3}}$. The limit does not exist.

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Note that $$\lim_{x\to2}\frac{\sqrt{1+\sqrt{x+2}-\sqrt3}}{x-2}$$ The deonominator goes to zero, but the numerator goes to $\sqrt{3-\sqrt{3}}$. Thus, we know that the limit does not exist.