Question

evaluate the above limit​

Answer 1

Answer

Using L'Hopital's rule:

Since evaluating the limits of the numerator and denominator it would be in a intermediate form

$\lim_{x→\pi}( \frac{ \frac{dx}{dy} ( \sqrt{1 - \cos(x) } )}{ \frac{dx}{dy}( { \sin(x) }^{2} ) } )$

Calculate the derivatives

$\lim_{x→\pi}( \frac{ \sin(x) }{4 \sqrt{1 - \cos(x) } \times \cos(x) } )$

$\lim_{x→\pi }( \frac{ \frac{ \sin(x) }{2 \sqrt{1 - \cos(x) } } }{ \sin(2x) } )$

$\lim_{x→\pi}( \frac{ \sin(x) }{2 \sqrt{1 - \cos(x) \times \sin(2x) } } )$

We know that sin(2x) = 2sin(x)cos(x)

$\lim_{x→\pi}( \frac{ \sin(x) }{2 \sqrt{1 - \cos(x) } \times 2 \sin(x) \cos(x) } )$

$\lim_{x→\pi}( \frac{1}{4\sqrt{1 - \cos(x) } \times \cos(x) } )$

Evaluate the limit

$= \frac{1}{4 \sqrt{1 - \cos(\pi) } \times \cos(\pi) }$

$= - \dfrac{ \sqrt{2} }{8}$