Evaluate the limit of $e^{\pi-\ln \frac{x+4}{-x}}/x$ as $x\to 0$

Question

Evaluate:$$\lim_{x \to \ 0^-}\frac{e^{\pi-\ln \frac{x+4}{-x}}}{x} $$

I tried Hopital's rule, even the Taylor series of the function $e^x$ without success. So how can one solve it?

Answer 1

$$e^{\pi-\ln\frac{x+4}{x}}=e^{\pi}.\dfrac{1}{\dfrac{x+4}{-x}}=e^{\pi}\dfrac{-x}{x+4}$$

Hence $$\lim_{x \to 0^{-}}\frac{e^{\pi-\ln\frac{x+4}{-x}}}{x}=\lim_{x \to 0^{-}}\frac{-e^{\pi}}{x+4}$$