An important application of determining the limit of a function is to understand the behavior of the graph of that function. The idea of a limit might have been introduced first to you in a college algebra class when talking about horizontal asymptotes. There is no horizontal asymptote.
Lets look at the three cases again using limits.
This time we will take the limit as x approaches infinity. When you are evaluating infinite limits first determine the highest power of x in the function. In this case, the highest power of x is to the third power. We can divide both the numerator and the denominator by that power. The limit is undefined.