# please explain this derivative problem?

Let f be a function such that

lim h-->0 f(1 + h) = 2 , and lim h-->0 (f(1 + h) − f(1))/h = 3 .

Which of the following statements are true?

A. f has a removable discontinuity at x = 1,

B. f is differentiable at x = 1 ,

C. f(1) = 3, f0(1) = 2 .

1. A only

2. A and B only

3. all are true

4. none are true

5. B only

6. B and C only

7. A and C only

8. C only

please, explain the problem to me.

Relevance
• 9 years ago

Hi,

from what you gave me, this is what I surmise:

if h -> 0 in the first definition you gave, then f(1) = 2.

based on the second, that is basically the derivative evaluated at x=1. they pretty much told you that it can be derived at 1 so B is certainly true.

If if can be derived at 1, it is continuous at 1 and we know that f(1) = 2 rather than 3 so A and C are false.

The correct answer is 5. b only. (I CANT READ IF YOU PUT F ' (1) = 3. IF YOU MEANT THE DERIVATIVE F ' (1) = 3 AND THAT THE ORIGINAL FUNCTION AT 1, F(1) = 2, THEN C IS ALSO CORRECT. PLEASE NOTE THIS. THE ANSWER WOULD BE 6, B AND C.)

• 9 years ago

Based on what you've provided...#5 is the only correct answer.

I don't know what f0(1) = 2 means. 1st condition f(1) = 2. , 2nd condition derivative exists and is differentiable at x = 1. but that is it. C is not a valid answer.

Source(s): Math tutor for several years.
• s k
Lv 7
9 years ago

B and C are true.

lim_{h->0} (f(1 + h) - f(1))/h = 3 means that f is differentiable at x = 1 and f'(1) = 3.

As f is differentiable at 1, it must also be continuous at 1. That is, f(1) = lim_{x->1} f(x).

lim_{x->1} f(x) = lim_{h->0} f(1 + h) = 2. So f(1) = 2.