Yes, fx[n_] is basically equivalent to Rationalize, albeit a bit different.

What about

max=40;
fx[n_] := Module[{a},
  a = Take[RealDigits[SetPrecision[N[Sqrt[2]], max]][[1]], 2 ;; n];
  1 + FromDigits[a]/10^Length[a]]

But does that really make sense? For these small differences your function will be a straight line.

Thanks Hans,

Yet it does not work for n >12... I need it to work for big n...20 or more...

Thanks for the hints guys. I think my question really can be put as: How can I Plot the graphic of f(x)=2^x, for instance, in the interval (N[Sqrt[2],n],N[Sqrt[2],n+1]) ? take n=10 for example...

Thanks Hans. Kind of it but I thought about using something like Manipulate to be able to zoom in near and near to be able to "see" the sequence of Exp[x_n] getting near and near the value of Exp[Sqrt[2]]

Explaining better: I want to give a sequence o rational numbers, xn converging to sqrt[2] and I want to be able to plot in a nice visible way the correponding sequence of yn=Exp[x_n]; Better yet a want to be able to plot yn as a function of xn in order to visualize where y_n is converging to.

I'm not sure exactly what kind of output you would like to get here. The rational numbers are dense in the reals, so you would be plotting your function for 14/10 and 15/10, but then also for the midpoint 29/20, and the midpoint between that and the boundaries, and the midpoints between midpoints. Even so, that would only give a subset of the rationals equivalent to the set of binary fractions.

I guess you could use ListPlot, with the argument of ListPlot being some set of rational numbers, maybe even ordering them by where they appear in the famous snake-like path through the rationals

1/1 2/1 3/1
1/2 2/2 3/2
1/3 2/3 3/3

Then the code would be

ListPlot[ Table[{x,Exp[x]}, {x, {1,2,1/2,1/3,2/3,3/2,3}}] ]

Do you have a more detailed description of what you have in mind? Maybe with an example?

Okay, if you specifically want, not just rational numbers, but successively better rational approximations to Sqrt[2], then you could use

Table[Rationalize[N[Sqrt[2], n], 0], {n, 1, 20}]

To break this down

N[Sqrt[2], n]

returns a floating-point approximation of Sqrt[2] to 'n' digits.

Rationalize[..., 0]

returns the rational number closest to the input, with '0' indicating the error tolerance.

Table[..., {n, 1, 20}]

returns the input for each value of 'n' from 1 to 20.

Tying it all together, we can put this list of rational numbers in as an argument to ListPlot:

vals=Table[Rationalize[N[Sqrt[2], n], 0], {n, 1, 10}];
ListPlot[
    Table[
        {x, Exp[x]}, 
        {x, vals}
    ]
]

If you're new to the syntax of the Wolfram Language, there is a wide range of free resources at https://support.wolfram.com/30451, and you can learn about each of the functions used here in the documentation center at https://reference.wolfram.com/language/ .