# Techniques for connecting two asymptotic behaviors to smooth curve fitting

I have a set of data points $f(x_i)$ for some instances $x_i$, $i\leq N$. Also, I was able to fit the curve asymptotically due to prior knowledge, i.e. I was able to fit the data to functional forms $f(x\ll x_0), f(x\gg x_0)$. How can I proceed in order to connect the two regions smoothly to fit the entire data points to a curve? Are there techniques to do this given $x_i, f(x_i), f(x\ll x_0), f(x\gg x_0)$?

• A little bit more info would help... e.g. what’s the behavior now of $\hat{f}$ around $x_0$ and what would you like it to be? – Jim Mar 11 '18 at 14:11
• As I understand your question, you would like to find a point where both equations produce the same value and also have the same first derivative. – James Phillips Mar 11 '18 at 20:49
• If you are able to use a programmatic technique, you can select a small range of overlap and smoothly vary from one to another. For example, if the overlap range is from 5.0 to 6.0, at 5.25 the result is 25% of the first equation's value plus 75% of the second equation value. At 5.50 the result would be 50% of each. You can plot the result against the data to verify that the overall program result meets your requirements. – James Phillips Mar 13 '18 at 10:10