# Verifying that an inequality has a finite limit

I am trying to verify the following inequality using Mathematica, but without any success.

$$\lim_{a\rightarrow0^{+}}\frac{\intop_{0}^{1}e^{\frac{-30}{\sqrt{a^{16}*\log\left(\frac{1}{x}\right)}}}*xdx}{e^{\frac{-32}{a}}}<+\infty.$$

Mathematica just retrieves the same expression after a couple of minutes.

• Have you tried numerics? The limit appears to be zero. – AccidentalFourierTransform Aug 24 at 17:13
• It is really strange to see $*$ to denote multiplication. * is used for this only in computer languages, not in textbook notation. $*$ normally means convolution or other uncommon or specialized operations. – Szabolcs Aug 24 at 17:36
• Please provide the actual code you tried. – Daniel Lichtblau Aug 25 at 18:10

If you do the integral first and then take the limit, you get an answer, but not the one you want.

int = Integrate[x/E^(30/(a^16*Log[1/x])^(1/15)), {x, 0, 1}]
(* (Sqrt*MeijerG[{{}, {}}, {{0, 1/15, 2/15, 1/5, 4/15, 1/3, 2/5,
7/15, 8/15, 3/5, 2/3, 11/15, 4/5, 13/15, 14/15, 1}, {}},
65536/a^16])/(256*Pi^7) *)


limabove=Limit[int/E^(-32/a), a -> 0, Direction -> "FromAbove"]


N[limabove, 50]//Chop
(* ∞ *)


The Chop gets rid of the small imaginary part due to numerical round off error.

If you evaluate in the other direction you get zero

Limit[int/E^(-32/a), a -> 0, Direction -> "FromBelow"]
(* 0 *)


but that does take awhile.

• I could not run the code: limabove=Limit[int/E^(-32/a), a -> 0, Direction -> "FromAbove"]. Mathematica says that: Limit::ldir: Value of Direction -> FromAbove should be a number or Automatic. >> Then I tried: limabove=Limit[int/E^(-32/a), a -> 0, Direction -> -1"], but it retrieves the same expression without calculating. – Hana Lee Aug 24 at 23:32
• The code works for Version 12. The options "FromAbove" and "FromBelow" are fairly new and may not be available if you are using an older Mathematica version. The directions +1 and -1 have been around awhile and should work in older versions, but an older version may not have as good limit finding capabilities as newer versions. You have not said what version you are using. – Bill Watts Aug 25 at 5:47
• I am using Mathematica 9. I should update it now. Thanks. – Hana Lee Aug 25 at 6:14
• Hi Bill, does Mathematica use background network to improve its computation? I did install Mathematica 12 and run the code you gave (internet off), but it took so long to evaluate the limit, now I still don't get the answer. – Hana Lee Aug 26 at 22:00
• Internet is not required to run and I don't think it speeds things up, but it did take me quite awhile to run and I have a pretty fast gaming computer. – Bill Watts Aug 26 at 23:46