TITLE: Find the limit of the complex function.
QUESTION [8 upvotes]: $$ \lim_{x\to a} \left(2- \frac{x}{a}\right)^{\left(\tan \frac{\pi x}{2a}\right)}.$$
I have simplified this limit to this extent :
$$e^{ \lim_{x\to a} \left(\left(1- \frac{x}{a}\right){\left(\tan \frac{\pi x}{2a}\right)}\right)}$$
I don't know how to simplify the limit after that. please help.

REPLY [2 votes]: Use a change of variable
write $$ \lim_{x\to a}(1 - x/a)\tan(\frac{\pi x}{ 2a}) = \lim_{x\to a} \frac{(1 - x/a)}{\cos(\frac{\pi x}{ 2a})}\sin(\frac{\pi x}{ 2a}) $$
To solve the indetermination make the change of variable: $ t = 1- \frac{x}{a}$. You have that $t \to 0$ when $x \to a$.
$$\lim_{t \to 0} \frac{t}{ \cos (\frac{\pi}{2}(1-t))} = \lim_{t \to 0} \frac{t}{ \cos(\frac{\pi}{2})\cos(\frac{\pi}{2}t) + \sin(\frac{\pi}{2})\sin (\frac{\pi}{2}t)} =  \lim_{t \to 0} \frac{t}{\sin (\frac{\pi}{2}t)}  = \frac{2}{\pi}$$
Por the last equation use that
$$\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)$$
$$ \frac{\sin(\nabla)}{\nabla} \to 1 \quad \text{ if } \quad \nabla \to 0 $$