\begin{align*} &\;\text{For fixed $x \in [-1,1]$,}\\\\[6pt] &\lim_{n \to \infty}x^{1+\frac{1}{2n-1}}\\\\[4pt] =\;&\lim_{n \to \infty}x^{\frac{2n}{2n-1}}\\\\[4pt] =\;&\lim_{n \to \infty}(x^2)^{\frac{n}{2n-1}}\\\\[4pt] =\;&\lim_{n \to \infty}(|x|^2)^{\frac{n}{2n-1}}\\\\[4pt] =\;&\lim_{n \to \infty}(|x|)^{\frac{2n}{2n-1}}\\\\[4pt] =\;&\lim_{n \to \infty}(|x|)^{1}\\\\[4pt] =\;&\lim_{n \to \infty}|x|\\\\[4pt] =\;&|x|\\\\[12pt] &\;\text{Alternatively, using the author's approach,}\\\\[6pt] &\lim_{n \to \infty}x^{1+\frac{1}{2n-1}}\\\\[4pt] =\;&x\lim_{n \to \infty}x^{\frac{1}{2n-1}}\\\\[4pt] =\;&x\bigl(\text{sgn}(x)\bigr)\qquad\text{[since odd roots retain the sign of the input]}\\\\[4pt] =\;&|x|\\\\[4pt] \end{align*} where $\text{sgn}(x)$ is the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$ \text{sgn}(x) = \begin{cases} -1&\text{if}\;x<0 \\\\[4pt] 0&\text{if}\;x=0\\\\[4pt] 1&\text{if}\;x>0\\\\[4pt] \end{cases} $$