Evaluate the following limit : lim(x→∞) √x{√(x + 1) - √x}

Question

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

lim(x→∞) √x{√(x + 1) - √x}

Answer 1

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

Now, Rationalizing the Numerator, we get

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