I noticed some unexpected behavior in the real-valued f(x)=(1+x)^1/x, as a function of real numbers, when plotting it on wolfram alpha. I inputed:(adsbygoogle = window.adsbygoogle || []).push({});

plot (1+x)^1/x from x=-0.0000001 to x=0.0000001

and saw that it unexpectedly seemed to oscillate near zero. I took a closer look with:

plot (1+x)^1/x from x=-0.00000000001 to x=0.00000000001

and saw that it definitely seems to oscillate near zero. My original rough graph on paper using a hand calculator suggested the curve was smooth near zero, and even windows calculator's 32 decimal places were unable to reveal the oscillation when I manually calculated many different values near zero.

I don't think f(x) is smooth near zero, though, since

d/dx[f(x)]=f(x)d/dx[1/xln(1+x)]

so all of f(x)'s derivatives have a discontinuity at zero. Since Leonhard Euler showed limf(x)=e as x approaches zero, and this limit is the same as you approach zero from the left or from the right, then the discontinuity of f(x) at zero is removable and the function

f(x)=(1+x)^1/x, x not = 0

f(0)=0 x=0

is continous. But the discontinuities in the derivatives don't appear to be removable. Each derivative will have a term containing the factor 1/x (not the 1/x in the exponent of f(x)) which cannot be removed. E.g.

d/dx[f(x)]=f(x)[(1/x)(1/(1+x)+(-1/x^2)(ln(x+1))]

the 1/x in the 1st term is not removable.

I'm not saying it doesn't oscillate near zero, I'm just saying that this is unexpected. I need confirming opinions and confirming logic to explain it so I can believe it.

So the questions are:

1. Does f(x) oscillate near zero?

2. Why does f(x) oscillate near zero?

3. Why does this behavior not appear until |x|<~10^-7?

4. Is it a glitch in Mathimatica? [doubt it]

5. If it doesn't oscillate [which it probably does] then how is the behavior of the derivatives explained near zero?

6. If it oscillates, then why is it so hard to replicate this oscillation manually on a calculator?

Note#1: This behavior concerns the real-valued f(x). I don't think Re[f(z)] or Im[f(z)], z complex, are of interest here.

Note#2: This behavior concerns (1+x)^1/x. I dont think either (1+1/x)^x or lim(1+1/x)^x as x goes to infinty are of interest here.

Thx in advance and Kudos to the person with the explanation!

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# (1+x)^1/x near zero

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