Como definir el concepto de límite a través de entornos porr faa?

<img src="https://tex.z-dn.net/?f=%5Ctexttt%7BSupongamos%20que%20el%20siguiente%20l%5C%27imite%20exista%20%7D%5Clim%5Climits_%7Bx%5Cto%20x_0%7Df%28x%29%3Da%5C%5C%20%5C%5C%5Ctexttt%7Bdonde%20%7Df%3A%5Cmathbb%7BR%7D%20%5Csupset%20X%5Cto%20%5Cmathbb%7BR%7D%5Ctexttt%7B%20entonces%20diremos%3A%20%7D%5C%5C%20%5C%5C%0A%5Chspace%2A%7B2.5cm%7D%5Cforall%20U_%5Cvarepsilon%28a%29%5C%2C%5Cexists%20U_%5Cdelta%28x_0%29%3A%20f%5Cleft%28U_%5Cdelta%28x_0%29%5Cright%29%5Csubset%20U_%5Cvarepsilon%28a%29%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctexttt%7BDonde%2C%20como%20es%20sabido%2C%20%7D%5Cvarepsilon%5C%20%5Ctextgreater%20%5C%200%5C%3B%2C%5C%3B%20%5Cdelta%5C%20%5Ctextgreater%20%5C%200%5C%5C%20%5C%5C%20%5C%5C%0A" />

Solo basta con recordar la definición siguiente

<img src="https://tex.z-dn.net/?f=%5Cboxed%7B%5Cboxed%7B%5Clim%5Climits_%7Bx%5Cto%20x_0%7Df%28x%29%3Da%5Cequiv%20%5Cforall%5Cvarepsilon%5C%20%5Ctextgreater%20%5C%200%2C%5Cexists%5Cdelta%5C%20%5Ctextgreater%20%5C%200%3A%20%7Cx-x_0%7C%5C%20%5Ctextless%20%5C%20%5Cdelta%5CLongrightarrow%20%7Cf%28x%29-a%7C%5C%20%5Ctextless%20%5C%20%5Cvarepsilon%7D%7D" />.