LIMEST will find the limit of a general function (specified only for evaluation) at a given point. You might think of limest like quad, but for limits.
While to me this seems to appear more often as a homework exercise than anything else, it was an interesting problem to solve as robustly as possible for a general case.
As an example, I'll use a moderately difficult one that is simple to analyze, but more difficult to deal with numerically.
fun = @(x) (exp(x)-1-x)./x.^2;
This function cannot be evaluated in MATLAB at x = 0, returning a NaN. While a Taylor series expansion shows the limit to be 1/2, the brute force evaluation of fun anywhere near zero results in numerical trash because of the two orders of cancellation.
Limest computes the limit, also returning an approximate error estimate.
[lim,err] = limest(fun,0)
I've now added the residueEst tool, for computation of the residue of a function at a known pole. For example, here is a function with a first order pole at z = 0
Again, both an estimate of the residue, as well as an uncertainty around that estimate are provided. Next, consider a function with a second order pole around z = pi.
See the included demos for many other examples of use.
John D'Errico (2022). Adaptive numerical limit (and residue) estimation (https://www.mathworks.com/matlabcentral/fileexchange/20058-adaptive-numerical-limit-and-residue-estimation), MATLAB Central File Exchange. Retrieved .
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Inspired by: Adaptive Robust Numerical Differentiation
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