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# Determine whether given statements about a function differentiable at a point are true or false

Let be a function and a point such that exists. Determine which of the following are true or false.

1. .
2. .
3. .
4. .

1. True. Let . Then as implies as . Thus,

2. True. Let , then

3. 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 .)

4. False. Here we evaluate,

### One comment

1. William says:

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)