Answer
$e^{1/2}$
Work Step by Step
Here,we have $\ln f(x)=\ln (1+2x)^{1/2 \ln x} $
and $\ln f(x)= \dfrac{\ln (1+2x)}{2 \ln x}$
But $\lim\limits_{x \to \infty} f(\infty)=\dfrac{\infty}{\infty}$
This shows an indeterminate form of limit, thus we will apply L-Hospital's rule such as:
$\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{p'(x)}{q'(x)}$
$e^{\lim\limits_{x \to \infty} \dfrac{2/1+2x}{2/ x}}=e^{\lim\limits_{x \to \infty} \frac{1}{1/(x+2)}}$
Thus,
$e^{\lim\limits_{x \to \infty} \dfrac{1}{1/ (x+2)}}=e^{1/2}$
