Definition of limits at infinity :

Let be a real number.

1. The statement means that for each there exists an such that whenever .

2. The statement means that for each there exists an such that whenever .

The limit expression is .

Here and .

.

, whenever .

If , then .

As  , then .

Let .

For , .

Thus, for each there exists an such that whenever .

Tha statement is proved.