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Evaluate the following limit : \(\lim\limits_{\text x \to \infty}\text x\{\sqrt{\text x+1}-\sqrt {\text x}\}\sqrt{\text x+2} \)

lim(x→∞) x{√(x+ 1) - √x} √(x + 2)

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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, \(\lim\limits_{\text x \to \infty}\text x\{\sqrt{\text x+1}-\sqrt {\text x}\}\sqrt{\text x+2} \) = \(\cfrac12\)

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