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)