$$\lim_{x \to \infty}\left(\frac{\left(\left(\frac{1}{4}\right)^{x}\right)!}{\left(\left(\frac{1}{5}\right)^{x}\right)!}\right) = 1$$
$$\lim_{x \to 0^-}\left(\frac{\left(\left(\frac{1}{4}\right)^{x}\right)!}{\left(\left(\frac{1}{5}\right)^{x}\right)!}\right) = 1$$
Más detalles con x→0 a la izquierda$$\lim_{x \to 0^+}\left(\frac{\left(\left(\frac{1}{4}\right)^{x}\right)!}{\left(\left(\frac{1}{5}\right)^{x}\right)!}\right) = 1$$
Más detalles con x→0 a la derecha$$\lim_{x \to 1^-}\left(\frac{\left(\left(\frac{1}{4}\right)^{x}\right)!}{\left(\left(\frac{1}{5}\right)^{x}\right)!}\right) = \frac{\Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{6}{5}\right)}$$
Más detalles con x→1 a la izquierda$$\lim_{x \to 1^+}\left(\frac{\left(\left(\frac{1}{4}\right)^{x}\right)!}{\left(\left(\frac{1}{5}\right)^{x}\right)!}\right) = \frac{\Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{6}{5}\right)}$$
Más detalles con x→1 a la derecha$$\lim_{x \to -\infty}\left(\frac{\left(\left(\frac{1}{4}\right)^{x}\right)!}{\left(\left(\frac{1}{5}\right)^{x}\right)!}\right)$$
Más detalles con x→-oo