F(x) = x + 1 sobre x2 - 2x - 3?

<img src="https://tex.z-dn.net/?f=%24f%28x%29%3D%5Cfrac%7Bx%2B1%7D%7Bx%5E2-2x-3%7D%24" />

<img src="https://tex.z-dn.net/?f=%24f%28x%29%3D%5Cfrac%7Bx%2B1%7D%7B%28x%2B1%29%28x-3%29%7D%24" />

<img src="https://tex.z-dn.net/?f=%24f%28x%29%3D%5Cfrac%7B1%7D%7Bx-3%7D%24" />

con la condición que no está definida en x = - 1 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Veamos si hay punto crítico

<img src="https://tex.z-dn.net/?f=%24f%27%28x%29%3D-%5Cfrac%7B1%7D%7B%28x-3%29%5E2%7D%24" />

es obvio que nunca es cero, y no hay punto crítico.

Además es

<img src="https://tex.z-dn.net/?f=f%27%28x%29%20%5C%20%5Ctextless%20%5C%200%5C%2C%20%2C%20%5Cforall%20x%20%5Cin%20%5Cmathbb%20R%20-%5C%7B1%2C3%5C%7D" />

es decir que es decreciente en todo su dominio = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

<img src="https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%5Cto%203%5E-%7Df%28x%29%3D-%5Cinfty%20%5Cwedge%20%5Clim%5Climits_%7Bx%5Cto%203%5E%2B%7Df%28x%29%3D%2B%5Cinfty" />

entonces x = 3 es una asíntota = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

<img src="https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%5Cto%5Cinfty%7Df%28x%29%3D0" />

o sea y = 0 es su asíntota horizontal = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =.