If $\lim _{x ightarrow 0}\left\{ an \left(\frac{\pi}{4}+x ight) ight\}^{1 / x}=e^n$, $ n=____?$

Question

If $\lim _{x ightarrow 0}\left\{	an \left(\frac{\pi}{4}+xight)ight\}^{1 / x}=e^n$, $ n=____?$

Professor PC Thomas Classes · Answer

$\lim _{x ightarrow 0}\left\{	an \left(\frac{\pi}{4}+xight)ight\}^{\frac{1}{x}}$ $$ =\lim _{x ightarrow 0}\left\{\frac{	an \frac{\pi}{4}+	an x}{1-	an \frac{\pi}{4} 	an x}ight\}^{\frac{1}{x}} $$ $\left\{ight.$ Since $\left.	an (A+B)=\frac{	an A+	an B}{1-	an A 	an B}ight\}$\ $=\lim _{x ightarrow 0}\left\{\frac{1+	an x}{1-	an x}ight\}^{\frac{1}{x}}$ since $	an \frac{\pi}{4}=1$\ $=\lim _{x ightarrow 0} \frac{\left[(1+	an x)^{1 / 	an x}ight]^{	an x / x}}{\left[(1-	an x)^{-1 / 	an x}ight]^{-	an x / x}}$\ $=\frac{e^{1}}{e^{-1}}=e^{2}$.\