I'm wondering if someone can just take a look at my working and tell me if I've done anything wrong/forgotten to answer some part. The question is as below:
The limits of the function as X approaches 0 from the left is 5, so in order to make this function continuous we assign a = 2 to make the function as X approaches 0 from the right also = 5.
For differentiability, I've done the below:
for lim (x+2)^2 + 1 (as a = 2) as x appproaches 0 the derivative will be (f(x) - f(0))/x-0 Which is equal to: lim(x->0+) ((x+2)^2 + 1 - 5)/x as we recall f(0) = 5 as above, we expand: (x^2 + 4x - 4 + 4) = x + 4 = 4
Therefore the derivative of this function from the right is 4.
for lim (2x+5) as x appproaches 0 the derivative will be (f(x) - f(0))/x-0 Recall f(0) = 5, therefore: lim(x->0-) ((2x+5)-5)/x = 2
Therefore this function is not differentiable.