Evaluate lim(x→2) (x^2-4)/(√(x + 2) - √(3x - 2))

Question

 Evaluate \(\lim\limits_{\text x \to2}\left(\cfrac{\text x^2-4}{\sqrt{\text x+2}-\sqrt{3\text x-2}}\right) \)

lim(x→2) (x²-4)/(√(x + 2) - √(3x - 2))

Answer 1

 To evaluate \(\lim\limits_{\text x \to2}\left(\cfrac{\text x^2-4}{\sqrt{\text x+2}-\sqrt{3\text x-2}}\right) \)

lim(x→2) (x²-4)/(√(x + 2) - √(3x - 2))

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

Thus, the value of \(\lim\limits_{\text x \to2}\left(\cfrac{\text x^2-4}{\sqrt{\text x+2}-\sqrt{3\text x-2}}\right) \) is -8.