$\displaystyle x \sin(\dfrac{5\pi}{x})$
$\displaystyle \infty \sin (\dfrac{5\pi}{\infty})$
$\displaystyle \infty \sin(0)$
$\displaystyle \infty(0)$ Indeterminate
$\displaystyle \dfrac{\dfrac{5 \pi}{x}}{\dfrac{1}{x}}$
$\displaystyle \dfrac{\dfrac{d}{dx} \dfrac{5 \pi}{x}}{\dfrac{d}{dx} \dfrac{1}{x}}$
$\displaystyle \dfrac{\cos(\dfrac{x(0) - 5\pi(1)}{x^{2}})}{\dfrac{x(0) - (1)(1)}{x^{2}}}$
$\displaystyle \dfrac{\cos(\dfrac{ - 5\pi}{x^{2}})}{\dfrac{- 1}{x^{2}}}$
$\displaystyle \dfrac{\cos(\dfrac{ - 5\pi}{(\infty)^{2}})}{\dfrac{- 1}{(\infty)^{2}}} = \dfrac{1}{0}$ ??? What is this