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,
I think another way to do part d would be to let k = a+t.
We have 1/2 times the regular limit function (with k’s instead of a’s), and if we note that v -> a as t -> 0 we get 1/2 f'(a), which is not equal to f'(a)