Evaluate the following limit : lim(x→∞) x{√(x^2 + 1) - √(x^2 - 1)}

Question

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

lim(x→∞) x{√(x² + 1) - √(x² - 1)}

Answer 1

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

On Rationalizing the Numerator we get,

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

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