If lim (h->0) ( f(3+h) - f(3) ) / h = 0, then which of the following must be true?

Question

lim      ( f(3+h) - f(3) ) / h = 0

h->0

 

• I : f has a derivative at x=3
• II : f is continuous at x=3
• III : f has a critical value at x=3

1. I Only
2. II Only
3. I and II
4. I and III
5. I, II, and III

Answer 1

Looks like f'(3)=0

 

Derivative is zero makes it a critical value.

Differential guarantees continuity  

So all three statements are true