Evaluate lim(x→0) (√(a + x) - √a)/(x √(a(a +x)))

Question

Evaluate \(\lim\limits_{\text x \to0}\left(\cfrac{\sqrt{a+\text x}-\sqrt a}{\text x\sqrt{a(a+\text x)}}\right) \)

lim(x→0) (√(a + x) - √a)/(x √(a(a +x)))

Answer 1

To evaluate \(\lim\limits_{\text x \to0}\left(\cfrac{\sqrt{a+\text x}-\sqrt a}{\text x\sqrt{a(a+\text x)}}\right) \)

lim(x→0) (√(a + x) - √a)/(x √(a(a +x)))

Formula used: L'Hospital's rule

Let f(x) and g(x) be two functions which are differentiable on an open interval I except at a point a where

This represents an indeterminate form. Thus applying L'Hospital's rule, we get