# Thread: Limit Question

1. ## Limit Question

Hey guys,

I'm having a REAL problem with the following limit

$\displaystyle \displaystyle \lim_{p\to\infty} \biggl(\frac{{| d |}^{np+p-1} |x|^{p-1}}{(p-1)!} \{(|x| + |\alpha_1|) \ldots (|x| + |\alpha_n|) \}^p\biggr)$

I've tried the squeeze rule, but an upper bound eludes me. I've only found the lower bound, which is just 0. I've tried to group together all of the powers of p, as follows:

$\displaystyle \displaystyle \lim_{p\to\infty} \biggl(\frac{{| d x|}^{p-1}}{(p-1)!} \{|d|^n(|x| + |\alpha_1|) \ldots (|x| + |\alpha_n|) \}^p\biggr)$

and maybe use the fact that

$\displaystyle \displaystyle \frac{{| d x|}^{p-1}}{(p-1)!}$

tends to zero as p tends to infinity, since the bottom half rises faster than the top half. However, I'm stuck because I don't know how to go about this formally, or if this is the right way to go about it. I'd be really grateful if help was provided, and possibly a solution.

Thank you very very much in advance.

HTale

2. This is what I'd do:

$\displaystyle \left\{d^n(x+a_1)\cdots(x+a_n)\right\}^p\leq C^p$

for some value of $\displaystyle C$

Likewise $\displaystyle (dx)^{p-1}C^p\leq K^p$ for some K and:

$\displaystyle \lim_{p\to\infty}\frac{K^p}{(p-1)!}\to 0\; \forall K\in\mathbb{R}^+$

3. Hey Shawsend, thanks a billion for your quick reply. But I have one question regarding this part:

$\displaystyle \left\{d^n(x+a_1)\cdots(x+a_n)\right\}^p\leq C^p$ for some value of $\displaystyle C$

Can I assume that the polynomial is an integer if it is given for some fixed x? I think also the question applies for the other part too. That is, when writing the solution out, should I state that this is for a fixed x?

HTale

EDIT: Also, how would I go about proving the final limit? Just a few pointers will do. Once again, thanks in advance!

4. Originally Posted by HTale
Hey Shawsend, thanks a billion for your quick reply. But I have one question regarding this part:

$\displaystyle \left\{d^n(x+a_1)\cdots(x+a_n)\right\}^p\leq C^p$ for some value of $\displaystyle C$

Can I assume that the polynomial is an integer if it is given for some fixed x? I think also the question applies for the other part too. That is, when writing the solution out, should I state that this is for a fixed x?

HTale

EDIT: Also, how would I go about proving the final limit? Just a few pointers will do. Once again, thanks in advance!
I treated it as for a fixed $\displaystyle x,d$ and fixed set of $\displaystyle a_n$, I could always find a C which would make the expression true and as far as pointers, I'm sorry, I don't feel I'm good enough in analysis to be giving pointers. I just stated what I'd do. Maybe the analysis experts in here can make it more rigorous.

5. Originally Posted by HTale
Hey guys,

I'm having a REAL problem with the following limit

$\displaystyle \displaystyle \lim_{p\to\infty} \biggl(\frac{{| d |}^{np+p-1} |x|^{p-1}}{(p-1)!} \{(|x| + |\alpha_1|) \ldots (|x| + |\alpha_n|) \}^p\biggr)$

I've tried the squeeze rule, but an upper bound eludes me. I've only found the lower bound, which is just 0. I've tried to group together all of the powers of p, as follows:

$\displaystyle \displaystyle \lim_{p\to\infty} \biggl(\frac{{| d x|}^{p-1}}{(p-1)!} \{|d|^n(|x| + |\alpha_1|) \ldots (|x| + |\alpha_n|) \}^p\biggr)$

and maybe use the fact that

$\displaystyle \displaystyle \frac{{| d x|}^{p-1}}{(p-1)!}$

tends to zero as p tends to infinity, since the bottom half rises faster than the top half. However, I'm stuck because I don't know how to go about this formally, or if this is the right way to go about it. I'd be really grateful if help was provided, and possibly a solution.

Thank you very very much in advance.

HTale
This is very easy indeed...assuming I am understanding it correctly...by that I mean I am assuming that $\displaystyle \left\{d,x,\alpha,\cdots,\alpha_n\right\}\in\mathb b{R}$

If so then similarly to the previous answer let $\displaystyle \prod_{k=1}^{n}\bigg[|x|-\alpha_k\bigg]=C\in\mathbb{R}$

[tex]\Therefore $\displaystyle \lim_{p\to\infty}\frac{|dx|^{p-1}\prod_{k=1}^{n}\bigg[|x|-\alpha_k\bigg]}{(p-1)!}=\lim_{p\to\infty}\frac{|dx|^{p-1}C^p}{(p-1)!}=\frac{1}{dx}\lim_{p\to\infty}\frac{\left(C|dx |\right)^p}{(p-1)!}$

Now using the fact that $\displaystyle p!\sim\sqrt{2\pi{p}}p^pe^{-p}$

We can rewrite this as

$\displaystyle \frac{1}{|dx|}\lim_{p\to\infty}\frac{p\left(|dx|C\ right)^P}{\sqrt{2\pi{p}}p^pe^{-p}}=\frac{1}{|dx|\sqrt{2\pi}}\lim_{p\to\infty}\fra c{\sqrt{p}\left(e|dx|C\right)^p}{p^p}$$\displaystyle =\frac{1}{|dx|\sqrt{2\pi}}\lim_{p\to\infty}\sqrt{p }\left(\frac{M}{p}\right)^p$$\displaystyle \quad\text{where }M=e|dx|C$ From there it should be pretty apparent the answer is zero

6. Originally Posted by Mathstud28
This is very easy indeed...assuming I am understanding it correctly...by that I mean I am assuming that $\displaystyle \left\{d,x,\alpha,\cdots,\alpha_n\right\}\in\mathb b{R}$

If so then similarly to the previous answer let $\displaystyle \prod_{k=1}^{n}\bigg[|x|-\alpha_k\bigg]=C\in\mathbb{R}$

[tex]\Therefore $\displaystyle \lim_{p\to\infty}\frac{|dx|^{p-1}\prod_{k=1}^{n}\bigg[|x|-\alpha_k\bigg]}{(p-1)!}=\lim_{p\to\infty}\frac{|dx|^{p-1}C^p}{(p-1)!}=\frac{1}{dx}\lim_{p\to\infty}\frac{\left(C|dx |\right)^p}{(p-1)!}$

Now using the fact that $\displaystyle p!\sim\sqrt{2\pi{p}}p^pe^{-p}$

We can rewrite this as

$\displaystyle \frac{1}{|dx|}\lim_{p\to\infty}\frac{p\left(|dx|C\ right)^P}{\sqrt{2\pi{p}}p^pe^{-p}}=\frac{1}{|dx|\sqrt{2\pi}}\lim_{p\to\infty}\fra c{\sqrt{p}\left(e|dx|C\right)^p}{p^p}$$\displaystyle =\frac{1}{|dx|\sqrt{2\pi}}\lim_{p\to\infty}\sqrt{p }\left(\frac{M}{p}\right)^p$$\displaystyle \quad\text{where }M=e|dx|C$ From there it should be pretty apparent the answer is zero

Aha, Stirlings approximation! That's quite ingenious - thanks man! But, we are dealing in the complex domain - does this change things?

7. Originally Posted by HTale
Aha, Stirlings approximation! That's quite ingenious - thanks man! But, we are dealing in the complex domain - does this change things?
It should not.