The attached image shows a plot between two quantities $R_{th}$ known as thermal resistance and $d$. As can be seen, the thermal resistance shows a decrease as the channel depth increases. I want to define an optimum $d$ beyond which any decrement in $R_{th}$ becomes insignificant. Specifically, is there any well-defined criteria or definition to do this? For example, is there any guideline like when decrement gets below $0.01 % $ percentage for a 0.5 mm increase in length?
But here $0.01$ percentage would be an arbitrarily chosen value. Are there mathematical guidelines to select these values ?
That very much depends on what you want to do. E.g. $a^{-x}$ is so that it decreases by a fixed fraction for each increase of $x$, so "decreases to 1% of the starting value" is the same $x_0..x_f$ for each starting value. Other functions are different.
If for illustration purposes, I'd select a finish point so that it is clear where it is going (plot asymptote in light!), but still is distinguishable from it.
If for computing something, perhaps the point where the difference from the starting value to the asymptote is almost achieved (95%, 99% of the way).
If you need precision (and say the integral to $\infty$), check out e.g. Hildebrand's "Short course on asymptotics", a lucid explanation of important techiques to derive approximations, use them and, most importantly, find out how far off the approximations. References many more detailed works.