is limx->0^- f(x) and limx->0^+ f(x) equal

Question

is limx->0^- f(x) and limx->0^+ f(x) equal, if they are equal does that mean that they are differentiable?

Answer 1

Hey there,

I think this question may need some clarification, but here's a start.

To be differentiable at a point, we need two things:

1.) The function has to be continuous at that point.

2.) The left hand and right hand derivatives have to be equal (ie, not a cusp, corner, etc)

I believe this question is getting at the 1st point.

That is, for a function to be continuous at a specific x value, we need:

1.) The function must be defined at that x,

2.) The limit exists at that x (That is, the limit from the left is equal to the limit from the right)

3.) The limit must equal the defined value ( lim f(c) = f(c))

So given the limited info in your question, if the limit from the left of zero and the right of zero are equal the function may not be differentiable.

This could happen if the function isn't defined at zero (think of a hole)

Or if it is defined, but we have a corner or a cusp ( think of the absolute value function)

Hope that gets you going in the right direction!