$\text{If the }\lim \limits_{x\rightarrow \infty }y\text{ exists, then the end behavior and the }\lim \limits_{x\rightarrow \infty }y\text{ are the same.}$

(This usually means that there is a horizontal asymptote.)

However, there is an exception when a function has two different horizontal asymptotes:

$\text{for example, }y =\text{ arctan}x\text{ whose }\lim \limits_{x\rightarrow \infty }y\neq \lim \limits_{x\rightarrow -\infty }y.$

$\text{If the limit does not exist, and }\lim \limits_{x\rightarrow \infty }y=\pm \infty ,\text{ then the end behavior still exists.}$

End behavior is a function that describes the shape of its growth towards $∞$ or $− ∞$.