3
$\begingroup$

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.

$\endgroup$
7
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.