Given: \(\lim\limits_{\text x \to \infty}\text x\{\sqrt{\text x+1}-\sqrt {\text x}\}\sqrt{\text x+2} \)
On Rationalizing the numerator we get,
Dividing the numerator and the denominator by √x, we get,
Hence, \(\lim\limits_{\text x \to \infty}\text x\{\sqrt{\text x+1}-\sqrt {\text x}\}\sqrt{\text x+2} \) = \(\cfrac12\)