How to evaluate limit of this function?

Question

$$ \lim_{x\to\infty}\left(\frac{x}{2x +1}\right)^x$$ Answer is $0$, or it's indeterminate? Can we use Second Remarkable Limit? Also, using L Hospitals rule is not permitted.

Answer 1

Hint:

$\Big(\frac{x}{2x+1}\Big)^x=\frac{1}{2^x}\Big(\frac{1}{1+\frac{1/2}{x}}\Big)^x$

$\Big(1+\frac{1/2}{x}\Big)^x\xrightarrow{x\rightarrow\infty}e^{1/2}$

$\frac{1}{2^x}\xrightarrow{x\rightarrow\infty}0$