pls explain?

Question

which of the following statements are true for every function f,defined on the set of all real no.s such that lim(x tends to 0) f(x)/x is a real no. L & f(0)=0 ??

1)f is differentiable at 0

2)L=0

3)lim (x tends to 0) f(x)=0

Answer 1

Your givens are saying two things:
1. f(0) = 0 which means the function is defined at 0 and when x=0 f(x) also equals 0. So the function passes through the point (0,0).
2. You are also given that as x approaches 0 the function has a limit. (But keep in mind that the limit is not necessarily 0.)

In terms of the questions.
1. The function may or may not be differentiable at x. For example the function y = absolute value of x meets both the givens yet it is not differentiable at x=0 because there is a corner there, and we cannot differentiate at corners. So item 1 is not true for all functions.
2. This is also not true for all functions. If you have a discontinuous function (let's say it looks like the line 5x+6 EXCEPT at x= zero there is a hole in the line AND there is a point at (0,0)) the value of f(0) will be 0 but the limit as x approaches 0 is 6. Item 2 is not true for all functions.
3. Item 3 is saying the same thing as item 2. Item 2 is saying the limit is zero and item 3 is saying the limit is zero. Item 3 is not true for all functions.

Answer 2

lim f(x)/x when x tends to 0 is 0/0 though u can use the Hopital so u have lim f(x)/x = lim df/dx when x tends to 0 which means lim df when x tends to 0 is equal to L so f is difrentiable

Answer 3

all of the statements 1), 2), and 3) must be true for the conditions you stated to be satisfied

Answer 4

All of them