It's not very clear to me what is what in the images you link to, so I'll ignore them in my answer.

The order of a polynomial defines how many inflection points the fitted function can have and so dividing two polynomials in a rational fit will also have a defined number of inflection points based on the two polynomials used. Inflection points are critical as they allow a polynomial to adapt to changes in slope within the data. Over a smaller range of data the number of changes of slope that the rpf must adapt to is smaller and so becomes more able to track even subtle changes in slope. However if you use too small a range you risk over-fitting the data. Overlapped chunking of the data is an option but must be considered carefully to balance the risks of over and under-fitting your data. The order of the polynomial should be appropriate to the number of features you expect in your data range.

This means that to answer your question 'So how to get the right fit and right parameters for any function.' - you need to deciede in advance what region you need to carry out your fit over. The smaller the region the better the adaption to subtle changes in slope but the smaller range of data you describe and extrapolation is a big NO, so you must fit over the full range you need to use. As with statistics the correct approach is to work out what you need before analysing rather than retrospectively trying to find the approach that gives you an answer that seems 'good'.

Thanks for your reply James.Its is very informative.Fyi, basically the data are vibration characteristics of a beam measured using the accelerometer sensor.

I was trying to calculate the FRF of the beam which is basically the fft fo the signal upon the force function fft.

the 3 frequency peaks of the fft data represent that the beam vibrated at 3 different modes.

Now I am fitting the rpf function over the three peaks to to get the characteristics of individual peak that is it m (mass), K(spring constant) and c (damping constant values)

But every range of fit gives me a different k and c values.(im very sensitive to range of frequency I select)

So I was just wondering what is the actual value of parameters and how to find them and automated statistical approach to get the range over which the fitting gives the exact values !!

I've found that converting a Taylor series that is a local fit to a curve to a ratio of two polynomials tends to extend the fit.  See Eric Ulm's article below to see where he uses rational functions to speed up a complex derivative's calculation.