How to evaluate this limit
$\lim_{x\rightarrow\infty}(1+\sin(x))^{x}$
I try in many forms but I cannot evaluate this limit; please help with tips please.
How to evaluate this limit $\lim_{x\rightarrow\infty}(1+\sin(x))^{x}$
3
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limits
1 Answers
7
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.