Let be a function and a point such that exists. Determine which of the following are true or false.
- .
- .
- .
- .
- True. Let . Then as implies as . Thus,
- True. Let , then
- False. First, from the definition of limit we can conclude
(by taking ). Then we have,
Thus, this is not true in general (since there are many function differentiable at a point such that the derivative at that point is nonzero). ( Note: This statement would be true were the denominator in the given equation instead of .)
- False. Here we evaluate,