Limit of a function raised to a power I was working with extraction of  non-electrolytic solutions and was sketching a mathematical formulae to find the limit of extracting a solvent by Nernst equation when I stumbled on this limit. 
$$\lim_{x\to\infty}\left(\frac{2x}{2x+1}\right)^x $$
Can somebody explain how to obtain this limit. This might be a silly question to put up here but I couldn't find the solution with my knowledge on limits.
P.S- from experimental evidence it is evident that a limit do exist. 

Let's try answering a different question first. 
Try finding 
$$
\lim_{x \to \infty} \left(\frac{2x+1}{2x} \right)^{2x}
= \lim_{x \to \infty} \left(1+\frac{1}{2x} \right)^{2x}
$$
Then see if you can transform the expression within the limit into the one you desire by applying some continuous functions. Then take the limit. 
