Question

How can a function have Limit, L, as x approaches a, while f (a) ‚ L?

Answer 1

if f(x) is defined such that f(a) is not L, this can happen. For example, consider the "greatest integer" or "floor" function, also known as the postage stamp function.

Using the postage stamp version, let the first-class rate be 10 cents for any weight up to one ounce, and 5 cents per ounce or part thereof after that.

f(0) = 0
f(x) = 10 for 0 < x <= 1
f(x) = 15 for 1 < x <= 2
...

Limit of f(x) is 10 as x -> 0+ but f(0) = 0
limit of f(x) = 15 as x -> 1+, but f(1) = 10

and so on.