Question

Is the statement "if the limit of f(x) as x approaches c is 0, then there must exist a number k such that f(k)<0.001" a true or false statement?

Answer 1

It is true.

Explanation:

If `lim_(xrarrc)f(x) = 0`, the for any positive `epsilon`, (like `0.001`) there is a positive `delta` such that for every `x` with `0 < abs(x-c) < delta` we have `abs(f(x)-0) < epsilon`.

There is not only one value of `k`, but for the `delta` that works for `epsilon = 0.001`, every `x` in `(c-delta, c+delta)` has `absx < 0.001`, so it has `f(x) < 0.001`