Find the limit
\[
\lim _{x \rightarrow \infty}(1+2 x)^{\frac{3}{2 \ln x}}
\]

Step 1 ：We are given the limit \(\lim _{x \rightarrow \infty}(1+2 x)^{\frac{3}{2 \ln x}}\)

Step 2 ：This limit is of the form 1^∞, which is an indeterminate form.

Step 3 ：We can use L'Hopital's rule to solve this. But before that, we need to convert the expression into a form that we can apply L'Hopital's rule.

Step 4 ：We can do this by taking the natural logarithm of the expression, which simplifies the exponent.

Step 5 ：Let's denote \(f = (2*x + 1)^{3/(2*\log(x))}\), then \(\ln(f) = \log((2*x + 1)^{3/(2*\log(x))})\)

Step 6 ：We differentiate the numerator and the denominator of the fraction inside the logarithm to apply L'Hopital's rule. The numerator becomes \(\frac{3}{(2*x + 1)*\log(x)} - \frac{3*\log(2*x + 1)}{2*x*\log(x)^2}\) and the denominator becomes 1.

Step 7 ：Applying L'Hopital's rule, we find that the limit of the fraction inside the logarithm as x approaches infinity is 0.

Step 8 ：Exponentiating this result, we find that the original limit is 1.

Step 9 ：Final Answer: The limit of the given expression as x approaches infinity is \(\boxed{1}\)