Numeric Function Limit

Numerical limit: lim x→a f(x)

The limit lim x→a f(x) = L tells you the value f(x) heads toward as x gets arbitrarily close to a from both sides. The numerical method just plugs in points x = a ± ε with ε getting smaller and smaller (say 0.1, then 0.01, then 0.001) and watches whether the outputs settle on something. Take f(x) = (x² - 1) / (x - 1) at x = 1: plug it in directly and you get the indeterminate form 0/0, but factor and it collapses to f(x) = x + 1, so lim x→1 = 2. When you land on 0/0 or ∞/∞, L'Hôpital's rule says lim f/g = lim f'/g' provided that right-hand limit exists. One warning: the numerical view can lie to you with oscillatory functions like sin(1/x) near 0, or when floating-point cancellation kicks in, so always back it up algebraically.

Applications

Limits sit underneath derivatives and integrals in Calculus I/II and drive continuity analysis. They show up in the asymptotic behaviour of algorithms (Big-O), in how ML loss functions converge during gradient descent, in numerical optimisation when you pick a step size near critical points, and in physics for things like instantaneous velocity or the electric field at a point.

FAQ

What if left and right limits differ? Then the two-sided limit simply does not exist. You see this a lot with step functions and at vertical asymptotes, which is why it pays to check each side on its own.

When should I use L'Hôpital instead of numerical? Reach for it whenever you hit 0/0 or ∞/∞ and the derivatives are easy enough to write down. L'Hôpital gives you the exact answer, whereas the numerical route only hints at one.

Does lim x→a f(x) = f(a) always? Only if f happens to be continuous at a. With a removable discontinuity the limit is still there, it just won't match f(a).