Let f:(0, ) R be a function given by f(x)=((8)/(7))^( 8x)/( 7x), & 0where a,b Z.

Question

Let $f:(0,\pi )	o R$ be a function given by $f(x)={(\frac{8}{7})}^{\frac{	an 8x}{	an 7x}}, \& 0where $a,b\in Z$. If $f$ is continuous at $x=\frac{\pi }{2}$, then $

Accepted Answer

$\underset{x	o \frac{{\pi }^{-}}{2}}{lim}f(x)=f(\frac{\pi }{2})=\underset{x	o \frac{{\pi }^{+}}{2}}{lim}f(x)x=\frac{\pi }{2}$

$\begin{gathered} \Rightarrow \, \underset{x	o \frac{{\pi }^{-}}{2}}{lim}{(\frac{8}{7})}^{(\frac{	an 8x}{	an 7x})}\, x=\frac{\pi }{2}-h \ \Rightarrow \, \underset{h	o 0}{lim}{(\frac{8}{7})}^{\frac{	an (4\pi -8h)}{	an (3\pi +\frac{\pi }{2}-7h)}}=\underset{h	o 0}{lim}{(\frac{8}{7})}^{\frac{	an (-8h)}{\cot (7h)}}={(\frac{8}{7})}^{0}=1 \ \Rightarrow \, a-8=1\Rightarrow a=9 \end{gathered}$

$\underset{x	o \frac{{\pi }^{+}}{2}}{lim}(1+|\cot x|{)}^{\frac{b}{a}|	an x|},x=\frac{\pi }{2}+h$

$\underset{h	o 0}{lim}(1-	an h{)}^{-\frac{b}{9}\cot h}=\underset{h	o 0}{lim}(1-	an h{)}^{-\frac{b}{9}\cot h}$

$\begin{gathered} =\underset{h	o 0}{lim}(1-	an h{)}^{(\frac{-1}{	an h})\cdot (-	an h)\cdot (\frac{-b}{9}\cot h)} \ ={e}^{\frac{b}{9}}=1\, \Rightarrow b=0 \ \Rightarrow {a}^{2}+{b}^{2}=81+0=81 \end{gathered}$

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