Evaluate the following limit : lim(x→π) (√ (5 + cos x) - 2)/(π - x)^2

Question

Evaluate the following limit : \(\lim\limits_{\text x \to\pi}\cfrac{\sqrt{5+cos\,\text x}-2}{(\pi-\text x)^2} \)

lim(x→π) (√ (5 + cos x) - 2)/(π - x)²

Answer 1

Put π - x = h

∴ x = π - h

As x → π, h → 0

Answer 2

Given, \(\lim\limits_{\text x \to\pi}\cfrac{\sqrt{5+cos\,\text x}-2}{(\pi-\text x)^2} \)

If x → π, then π – x → 0, let π – x = y

Rationalize the Numerator

Hence, \(\lim\limits_{\text x \to\pi}\cfrac{\sqrt{5+cos\,\text x}-2}{(\pi-\text x)^2} \) = \(\cfrac18\)