I am looking at the following:
- Show that if $p,q$ are real polynomial of the same degree, then $p(x)\asymp q(x)$ as $x\rightarrow \pm \infty$.
- Show that the funcions $n^{-x}, n\in \mathbb{N}$ define an asymptotic sequence as $x\rightarrow +\infty$.
- Show that for a big enough $X>0$ for each $n\in \mathbb{N}$ there is a constant $C_n>0$ such that $$\left |\int_{\mathbb{R}}\text{exp}\left (iXt-t^2\right )\, dt\right |\leq \frac{C_n}{X^n}$$
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I have done the following:
The limit of a polynomial at infinity is equal to $\pm \infty$, it depends on the leading coefficient, right?
Therefore both limits, of $p$ and $q$ will be $\pm \infty$ and so it holds that $p(x)\approx q(x)$.
To show that it is an asymptotic sequence do we have to show that it converges?
Could you give me a hint? I don't really have an idea.
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EDIT:
As for (1):
Let's consider the definition for $p(x)\asymp q(x)$: there are $0<A<B$ such that $A|q(x)|\leq |p(x)|\leq B|q(x)|$ for $x\rightarrow \infty$.
To show the first inequality:
Do we consider the limit $\lim\inf_{|x|\rightarrow \infty}\frac{|p(x)|}{|q(x)|}$ ?
Since $\deg p=\deg q=:n$ it follows that the limit is equal to the fraction of the coefficients of $x^n$, correct?
We want to show that this limit is positive, don't we? This follows since we have the absolute value.
Is that correct?
For the second inequality:
Do we consider the limit $\lim\sup_{|x|\rightarrow \infty}\frac{|p(x)|}{|q(x)|}$ ?
Since $\deg p=\deg q=:n$ it follows that the limit is equal to the fraction of the coefficients of $x^n$.
We want to show that this limit is finite ($<\infty$), don't we? This follows since we have a fraction of two real positive constants.
Is that correct?