$P$ is a monic polynomial of degree $n$ , then which are correct?

Solution 1:

You need to use the intermediate value theorem:
1) Since $n$ is even, $\lim\limits_{x\to-\infty} P=\infty$. From that, we know that $\lim\limits_{x\to-\infty}\frac{P(x)}{e^x}=\infty$. We also know that $\lim\limits_{x\to\infty}\frac{P(x)}{e^x}=0$. Therefore, there is a value $x_0$ verifying $\frac{P(x_0)}{e^{x_0}}=K$ (Since $0<K$).

2) Since $n$ is odd, $\lim\limits_{x\to-\infty} P=-\infty$. From that, we know that $\lim\limits_{x\to-\infty}\frac{P(x)}{e^x}=-\infty$. We also know that $\lim\limits_{x\to\infty}\frac{P(x)}{e^x}=0$. Therefore, there is a value $x_0$ verifying $\frac{P(x_0)}{e^{x_0}}=K$ (Since $0>K$).

3) Your justification seems to be false, since $\lim\limits_{x_0\to\infty} e^{-x_0}P(x_0)=0^+$. Instead consider $K=\frac{1}{2}$, and $P(x)=x-1$.

4) Consider $K=1$, and $P(x)=x-1$.