Question

CLOSED QUESTION

Answer 1

\[\lim_{x \rightarrow \frac{ \pi }{ 2}}\frac{ \left( x-\frac{ \pi }{ 2 } \right)^2 }{ 1-\sin x }\]
\[(put~x=\frac{ \pi }{ 2 }+h,h \rightarrow0~as~x \rightarrow \frac{ \pi }{ 2 })\]
\[=\lim_{h \rightarrow 0}\frac{ \left( \frac{ \pi }{ 2 } +h-\frac{ \pi }{ 2 }\right)^2 }{ 1-\sin \left( \frac{ \pi }{ 2 } +h\right) }\]
\[=\lim_{h \rightarrow 0}\frac{ h^2 }{ 1-\cos h }\]
\[=\lim_{h \rightarrow 0}\frac{ h^2 }{ 2 \sin ^2 \frac{ h }{ 2 } }\]
\[=\frac{ 4 }{ 2 }\lim_{h \rightarrow 0}\frac{ \frac{ h^2 }{ 4 } }{ \sin ^2\frac{ h }{ 2 } }\]
\[=2\lim_{h \rightarrow 0}\left( \frac{ \frac{ h }{ 2 } }{ \sin \frac{ h }{ 2 } } \right)^2=2\times 1^2=2\]