Testing an identity on the computer

You will write a tester for identities. It measures the largest gap between two functions on an interval, decides whether they agree, and finds the widest interval around $0$ from a list on which they agree. The checker uses it on three identities from the lesson.

The idea. An identity is true on an interval when the two sides are equal at every point. A computer cannot try every point, but a fine grid of points shows a failure at once. The gap between two functions at a point is the absolute value of their difference.

Your tasks.

1. Finish max_gap(f, g, xs). Return the largest value of abs(f(x) - g(x)) over the numbers in xs.
2. Finish holds(f, g, a, b). Use $2001$ equally spaced points from a to b. Return True if the largest gap is below 1e-9.
3. Finish largest_k(f, g, ks). Go through the numbers k in the list ks. Return the largest k for which the identity holds on $[-k, k]$, or None if there is none.