The limit of a power as it tends to infinity.

[HTale]

I've come across a snag in a proof, and I've become a little exasperated by the following limit:

\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 \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 \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

 

[Dick]

There's nothing very delicate about that limit. C^(Kp)/p! goes to zero for any choice of C or K. Try to change it into a form like that. Start by picking M>|x|,|d|,|a1|,|a2|,...|an|.