The value of $k$, for which the function $f(x)= \begin{cases}\left(\frac{4}{5} ight)^{\frac{ an 4 x}{ an 5 x}} & , 0 \lt x \lt \frac{\pi}{2} \ \mathrm{k}+\frac{2}{5} & , x=\frac{\pi}{2}\end{cases}$ is continuous at $x=\frac{\pi}{2}$, is

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The value of $k$, for which the function $f(x)= \begin{cases}\left(\frac{4}{5}ight)^{\frac{	an 4 x}{	an 5 x}} & , 0 \lt x \lt \frac{\pi}{2} \ \mathrm{k}+\frac{2}{5} & , x=\frac{\pi}{2}\end{cases}$ is continuous at $x=\frac{\pi}{2}$, is

Aicharya · Accepted Answer

Since $\mathrm{f}(x)$ is continuous at $x=\frac{\pi}{2}$. $\begin{aligned} & 	herefore \quad \mathrm{f}\left(\frac{\pi}{2}ight)=\lim _{x ightarrow \frac{\pi}{2}}\left(\frac{4}{5}ight)^{\frac{	an 4 x}{	an 5 x}} \ & \quad \Rightarrow \mathrm{k}+\frac{2}{5}=\lim _{x ightarrow \frac{\pi}{2}}\left(\frac{4}{5}ight)^{\lim _{x ightarrow \frac{\pi}{2}}(	an 4 x \cot 5 x)} \ & \quad \Rightarrow \mathrm{k}+\frac{2}{5}=\left(\frac{4}{5}ight)^0=1 \ & \quad \Rightarrow \mathrm{k}=\frac{3}{5} \end{aligned}$

The value of $k$, for which the function $f(x)= \begin{cases}\left(\frac{4}{5}\right)^{\frac{\tan 4 x}{\tan…

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