Question

plz help to solve this​

Answer 1

Step-by-step explanation:

We have,

$= \lim_{x \rarr\pi} \frac{ \sqrt{5 + \cos(x) } - 2}{(\pi - x)^{2} }$

Let x - π = t, then we have,

$= \lim_{ y\rarr0} \frac{ \sqrt{5 + \cos(y + \pi) } - 2 }{ {y}^{2} }$

Putting y = x - π,

$= \lim_{ y\rarr0} \frac{ \sqrt{5 - \cos(y) } - 2 }{ {y}^{2} }$

$= \lim_{ y\rarr0} \frac{( \sqrt{5 + \cos(y) } - 2)( \sqrt{5 - \cos(y) } + 2) }{ {y}^{2}( \sqrt{5 - \cos(y) } + 2) }$

$= \lim_{ y\rarr0} \frac{1 + \cos(y) }{ {y}^{2}( \sqrt{5 - \cos(y) } + 2) }$

$= \lim_{ y\rarr0} \frac{1 + \cos(y) }{ {y}^{2} } .\lim_{ y\rarr0} \frac{1}{( \sqrt{5 - \cos(y) } + 2)}$

$= \lim_{ y\rarr0} \frac{ - \sin(y) }{2y} .\frac{1}{ \sqrt{5 - 1} + 2}$

$= \frac{1}{4} .\lim_{ y\rarr0} \frac{ - \cos(y) }{2}$

$= \frac{1}{4} . \frac{ - 1}{2}$

$= - \frac{1}{8}$