how to evaluate $\lim_{x \to \infty}(1+4/x)^\sqrt{x^2+1}$

$$ \lim_{x \to \infty}(1+4/x)^\sqrt{x^2+1} $$ is like¢ $$ \lim_{x \to \infty}(1+1/x)^x = e $$

I have actually changed $\sqrt{x^2+1}$ by $x$ yet I have not obtained the anticipated outcome ($e^4$).