Question

question is in the photo​

Answer 1

Answer:

$let, y=\lim_{x \to a}(2-x/a)tan\frac{\pi x}{2a}$

taking log on both sides

log(y)=$\lim_{x \to a}tanlog(\frac{\pi x}{2a} (2-x/a))$

        =$\lim_{x \to a}\frac{(log(2-x/a))}{cot\frac{\pi x}{2a} }$        [0/0 form]

applying L' Hospital rule

        =$\lim_{x \to a}\frac{\frac{1}{(2-x/a)}*(\frac{-1}{a}) }{-cosec^{2}\frac{\pi x}{2a} *\frac{\pi }{2a} }$

        =$\frac{(-1/a)}{(-\pi /2a)}$

        =$2/\pi$

∴log(y)=2/π

or,y=$e^{\frac{2}{\pi } }$

∴$\lim_{x \to a} (2-x/a)tan\frac{\pi x}{2a} =e^{2/\pi }$