Limit of a polynomial

[quasar987]

Question: How to show that the limit as x goes to infinity of a given polynomial diverge? We have

\lim_{x \rightarrow \infty} a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0

but cannot say the this limit is the sum of the limit of each term separetly because none of these limit exist. We cannot do this either

\lim_{x \rightarrow \infty} a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0 = \lim_{x \rightarrow \infty} x^n(a_n + \frac{a_{n-1}}{x}+...+\frac{a_1}{x^{n-1}}+\frac{a_0}{x^n})


and say the the limit is the product of the limit because one of the limit does not exist. So what then?


Thank you.

[arildno]

Let's take the case a_{n}>0
What you CAN show, using your last equation, is that there exist N_{0} so that \frac{a_{0}}{x^{n}}\geq{-\frac{a_{n}}{2n}} whenever x\geq{N}_{0}
Similarly, it exists N_{1} so that:
\frac{a_{1}}{x^{n-1}}\geq{-\frac{a_{n}}{2n}} whenever x\geq{N}_{1}
And so on.
Setting N equal to the maximum of these N_{i} values, yields the inequality:
x^{n}(a_{n}+++\frac{a_{0}}{x^{n}})\geq\frac{x^{n}a _{n}}{2}, x\geq{N}