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  • Let f(x)=(k cos⁡x)/(π-2x) when x not equal to pi/2 and if ,lim f(x) = f(pi/2)find the value of k.

Let f \left( x \right) =\frac{k\cos x}{ \pi -2x}   when x \neq \frac{\pi}{2}and 

f(x)=3  if \lim _{x \rightarrow \frac{\pi}{2}} f(x)=f\left(\frac{\pi}{2}\right) find the value of k.

 

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\\ LHL=\mathop{\lim }_{x \rightarrow \mathop{\frac{ \pi }{2}}^{-}} \left( \frac{k\cos \left( \frac{ \pi }{2}-h \right) }{ \pi -2 \left( \frac{ \pi }{2}-h \right) } \right) =\mathop{\lim }_{h \rightarrow \mathop{0}^{-}} \left( \frac{k\cos \left( \frac{ \pi }{2}-h \right) }{2h} \right) \\ \\ =\mathop{\lim }_{h \rightarrow \mathop{0}^{-}} \left( \frac{k\sin h}{2h} \right) =\frac{k}{2} \\ \\ RHL=\mathop{\lim }_{x \rightarrow \mathop{\frac{ \pi }{2}}^{+}} \left( \frac{k\cos \left( \frac{ \pi }{2}+h \right) }{ \pi -2 \left( \frac{ \pi }{2}+h \right) } \right) =\mathop{\lim }_{h \rightarrow \mathop{0}^{+}} \left( \frac{k\cos \left( \frac{ \pi }{2}+h \right) }{-2h} \right) \\ \\
    \\=\mathop{\lim }_{h \rightarrow \mathop{0}^{-}} \left( \frac{-k\sin h}{-2h} \right) =\frac{k}{2} \\ \\ \mathop{\lim }_{x \rightarrow \frac{ \pi }{2}}f \left( x \right) =\frac{k}{2}=3 \\ \\ k=6 \\ \\

 

 

 

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