\[ \begin{split} &\lim_{x \to \infty} \left(1+\frac{1}{x^2}\right)^{2x-1}=\\[6pt] &=\lim_{x \to \infty} \left(1+\frac{1}{x^2}\right)^{\dfrac{2x^2-x}{x}}=\\[6pt] &=\lim_{x \to \infty} \frac{\left[\left(1+\dfrac{1}{x^2}\right)^{x^2}\right]^{\dfrac{2}{x}}}{\left(1+\dfrac{1}{x^2}\right)^1}=\\[6pt] &=\lim_{x \to \infty} e^{\dfrac{2}{x}}=\\[6pt] &=e^0=1 \end{split} \]