please solve this and give the instructions too.... lim (2 - a/x)^tan(pi*x/2a) x->a

Raghu menon

please solve this and give the instructions too....lim (2 - a/x)^tan(pi*x/2a)x->a

Srujana

Let $L=$ $Lt_{x \rightarrow a}\ {\left(2-\frac{a}{x}\right)^{tan\left(\frac{\pi x}{2a}\right)}$

$ln L=Lt_{x \rightarrow a}\ {tan\left(\frac{\pi x}{2a}\right)}ln{\left(2-\frac{a}{x}\right)}$
$=Lt_{x \rightarrow a}\ \frac{ln{\left(2-\frac{a}{x}\right)}}{cot\left(\frac{\pi x}{2a}\right)}$

Using L'Hospital's rule

$ln L=Lt_{x \rightarrow a}\ \frac{-\frac{1}{a\left(2-\frac{a}{x}\right)}}{\frac{- \pi}{2a}csc^2\left(\frac{\pi x}{2a}\right)}$

$\Rightarrow L=e^{\frac{2}{\pi}}$

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