# Calculating a limit as parameter goes to infinity

I have a fair background in pure mathematics and right now my project is a numerical implementation of a certain algorithm. I have some numerical background, but not all that much, so the question is simple:

In my algorithm, I have to calculate the value of certain parameter $$h_r = \lim_{t \to \infty} I(t,r),$$ which we have proven exists. Calculating the expression $I$ is computationally fairly expensive, as it involves solving a non-linear PDE. The parameter $r$ indexes a suitably dense set of directions (unit vectors) and all the calculations have to be done for all of them.

How do I calculate the limit? I can guess at a few options, but I don't know what are the best practices, or if I am missing some elegant solution that is common knowledge.

1. Select a large enough $t$, heuristically or by experiment. This seems to be a bad idea except for a test to see if the algorithm as a whole works.
2. Use an increasing sequence of $t$ with some stopping condition, such as $|I(t_j,r) - I(t_{j+1},r)|$ (or, better yet, relative change) being small.
3. Calculate how large $t$ needs to be, in order for the error of $I$ to be sufficiently small. This might turn out to be very difficult, given the complicated nature of $I$.
• Please edits the tags and the title to improve the question. Jun 12 '17 at 12:34

Plot $I$ versus $1/t$. Fit a smooth curve and find the intercept. This works a lot better if you have some theory predicting the behavior of $I$. For example, an error analysis might give $$I(t)=I_\infty+\frac{C}{t^2}+\cdots$$ If you have two different values of $t$, you can write two equations and solve them to find the unknowns $I_\infty$ and $C$. With more values, plot $I(t)$ against $t^{-2}$. Does it look like a straight line? Or only if $t$ is big enough. Then fit a straight line to the values that appear to lie on one.