Why is the solution to $(1+\frac{1}{x})^{x+1}=x$ close to pi?

$$\left(1+\frac{1}{x}\right)^{x+1}=x$$

Solving for $x$ yields $x \approx 3.1410415254107885010$ which is close to $\pi$

Why is this so? The only plausible reason I found is because $\ln{\pi} \approx 1+\frac{1}{2\pi}$

Any other reason why?