Limit of $h_n(x)=x^{1+\frac{1}{2n-1}}$
We have
$$\large x^{\large 1+\frac{1}{2n-1}}=x^{\large\frac{2n}{2n-1}}=\left(x^2\right)^{\large\frac{n}{2n-1}}\xrightarrow{n\to\infty}\sqrt{x^2}=\lvert x\rvert$$
Limit of $h_n(x)=x^{1+\frac{1}{2n-1}}$
We have
$$\large x^{\large 1+\frac{1}{2n-1}}=x^{\large\frac{2n}{2n-1}}=\left(x^2\right)^{\large\frac{n}{2n-1}}\xrightarrow{n\to\infty}\sqrt{x^2}=\lvert x\rvert$$