TITLE: Plotting $\left(1+\frac{1}{x^n}\right)^{x^n}$.
QUESTION [15 upvotes]: When I plot the following function, the graph behaves strangely:
$$f(x) = \left(1+\frac{1}{x^{16}}\right)^{x^{16}}$$
While $\lim_{x\to +\infty} f(x) = e$ the graph starts to fade at $x \approx 6$. What's going on here? (plotted on my trusty old 32 bit PC.)

I guess it's because of computer approximation and loss of significant digits. So I started calculating the binary representation to see if this is the case. However in my calculations values of $x=8$ should still behave nicely.
If computer approximation would be the problem, then plotting this function on a 64 bit pc should evade the problem (a bit). I tried the Wolfram Alpha servers:

The problem remains for the same values of $x$.
Questions

Could someone pinpoint the problem? What about the 32 vs 64 bit plot?
Is there a way to predict at which $x$-value the graph of the function below would start to fail?
$$f_n(x) = \left(1+\frac{1}{x^n}\right)^{x^n}$$

REPLY [3 votes]: Here is a plot using the (arbitrary precision) calculator Pari/GP. I use 200 dec digits precision as default in my computations and got this plot without oscillation up to x=20:            

I tried it so far up to x=256; no oscillation. See here the image up to x=128 (just to have the left increase visible)