Give an example of a function f which is defined at c which limit as x approaches c of f(x) exists, but that limit does not equal f(c).
Certainly! One example of a function that satisfies the given condition is:
f(x) = { x, if x ≠ 1
2, if x = 1 }
In this case, let's consider c = 1. The limit of f(x) as x approaches 1 is equal to 2 because, for any x value close to 1 (but not equal to 1), the function f(x) takes on the value of x. However, when x = 1, f(x) is equal to 2, which is different from the limit.
To verify this, we can calculate the limit expression:
lim(x→1) f(x) = lim(x→1) x = 1
So, the limit of f(x) as x approaches 1 is indeed 1, while f(1) is equal to 2. Thus, the limit of f(x) at c = 1 exists, but it is not equal to f(c).