Mathematics

This is a limit problem involving nested radicals, where direct substitution gives the indeterminate form $\frac{0}{0}$. The key insight is to rationalize the numerator by multiplying by the conjugate $\sqrt{1+\sqrt{2+x}}+\sqrt{3}$, which will eliminate the outer square root and reveal the behavior near $x=2$. You'll then need to rationalize again to handle the inner radical, but this two-step approach transforms a messy nested expression into something you can evaluate.

This is a limit problem involving nested radicals, where direct substitution gives the indeterminate form $\frac{0}{0}$. The key insight is to rationalize the numerator by multiplying by the conjugate $\sqrt{1+\sqrt{2+x}}+\sqrt{3}$, which will eliminate the outer square root and reveal the behavior near $x=2$. You'll then need to rationalize again to handle the inner radical, but this two-step approach transforms a messy nested expression into something you can evaluate.