How find this limit $lim_{xto+infty}frac{e^{-2x}(cos{x}+2sin x)+e^{-x^2}(sin{x})^2}{e^{-x}(cos{x}+sin{x})}$

Find this limit.

$$\lim_{x\to+\infty}\dfrac{e^{-2x}(\cos{x}+2\sin x)+e^{-x^2}(\sin{x})^2}{e^{-x}(\cos{x}+\sin{x})}$$

This wolf can't have reslut. link

maybe this limit is not exsit? so how prove it?
Thank you

Answer

There are arbitrarily large $x$ at which our function is not even defined. And by taking $x$ close enough to $2n\pi+\frac{3\pi}{4}$, we can make our function arbitrarily large positive or negative.