Question

Let f be a function such that lim as h approaches 0 f(2+h)-f(2)/h=0 Which of the following must be true
1. f is continuous at x=2
2. f is differnetiable at x=2
3. f has a horizontal tangent line at x=2
4. f travels through the origin

1 2 and 3 are correct but how should I explain why

Answer 1

the limit exists, and it is in fact the derivative

Answer 2

2: This is a limit definition of a derivative, specifically of f(x) for x=2. The limit is definable as 0, so f'(2)=0; that is, f is differentiable at x=2.
1: A function can only be differentiable at a point if it is continuous at that point, so f is continuous at x=2.
3: f'(2)=0, which is a horizontal tangent (the slope of f is 0, or horizontal).

4: Incorrect as there is no indication that f passes through (0, 0).

Hope that helps!