# How to evaluate this limit $\lim_{x\rightarrow\infty}(1+\sin(x))^{x}$

How to evaluate this limit

$\lim_{x\rightarrow\infty}(1+\sin(x))^{x}$

Notice that $\sin(x)$ oscillates between $-1$ and $1$. So $1 + \sin(x)$ oscillates between $0$ and $2$. In particular, if $x = 2\pi n + \frac{\pi}{2}$, then $1 + \sin(x) = 2$, while if $x = 2\pi n + \frac{3\pi}{2}$ then $1 + \sin(x) = 0$. So the function in this limit is sometimes $0^x = 0$ and sometimes $2^x$. Since these don't go to the same value, the limit does not exist.