I have the following task I cannot solve:
Let p be a polynomial of the form $p(x)= x^{n}+\sum \limits_{k=0}^{n-1}a_{k}x^{k}.$
Prove that:
i) $\lim\limits_{x\to\infty}p(x)=+∞$
ii) $\lim\limits_{x\to-\infty}p(x)=+∞$ if n is even
iii) $\lim\limits_{x\to-\infty}p(x)=-∞$ if n is not even
Earlier I had to prove, that the following is true:
"Let $(x_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathbb{R}$, with $\lim{x\to\infty}x_{n}=+\infty$, and let $(y_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathbb{R}$ with $\lim\limits_{x\to\infty}y_{n}=0$.
Then: $\lim\limits_{n\to\infty}x_{n}y_{n}=sign(c)*\infty$"
So I think we need to use it. Remember: I have already proven the theorem. It is very important for ii) and iii).
In general I do not know what I have to do now. I'm a little unsure how exactly the steps need to be. Would someone be kind enough to give tips or help me? I am very grateful for that.