Hello,
How to solve the following question?
$\displaystyle lim_{x\rightarrow1^{-}}(2+cos(\pi x))^{tg(\frac{\pi x}{2})}$
Thanks in advance.
$\displaystyle \displaystyle\lim_{x\nearrow 1}(2+\cos\pi x)^{\tan\frac{\pi x}{2}}=\lim_{x\nearrow 1}\left[(1+1+\cos\pi x)^{\displaystyle\frac{1}{1+\cos\pi x}}\right]^{\displaystyle(1+\cos\pi x)\tan\frac{\pi x}{2}}=$
$\displaystyle \displaystyle=e^{\displaystyle\lim_{x\nearrow 1}2\cos^2\frac{\pi x}{2}\tan\frac{\pi x}{2}}=e^{\displaystyle\lim_{x\nearrow 1}\sin\pi x}=e^0=1$