Aren't the side derivatives the same for f(x)={x,x<0 & x+2,x<=0? Then isn't it diff at x=0?

Question

calculus, differentiability

Answer 1

 f(x) = x, for x < 0 and f(x) = x+2, for x≥0

 

lim_x→0^-  f(x) = lim_x→0^- (x) = 0

 

lim_x→0⁺  f(x) = lim_x→0⁺  (x+2) = 2

 

Since the one-sided limits are unequal, lim_x→0  f(x) does not exist.

 

So, f(x) is not continuous when x = 0.

 

Therefore, f'(0) does not exist.  

 

Answer 2

The derivative is the slope of the tangent line.  Since you have two linear functions, their derivatives will be constants.

 

At x≤0 and x=0, the derivative is positive.  That is because the slope of the original function is positive.  Since the derivative of a linear function eliminates the x variable, you are just left with the constant slope of the original function.