Question

Determine whether the function is differentiable at x=2

x^2+1 for x<(or equal to) 2
4x-3 for x>2

I did the math for the limits of both equations and they both approach to 4. So that means they are differentiable right?

Answer 1

Any ideas?

Answer 2

for the function to be continuous, the limit on both sides must be the same. They are both 5 (not 4!).

So, your function is continuous. But that is not enough. Think of f(x) = |x|. It is continuous, but not differentiable at x=0.

For it to be differentiable, the derivative on both sides must exist and have the same limit. For your function, the derivatives are

left: 2x
right: 4
2x=4 at x=2, so it is differentiable. That means it is in some sense "smooth" where the pieces meet:

http://www.wolframalpha.com/input/?i=plot+y%3Dx%5E2%2B1,+y%3D4x-3,+0+%3C%3D+x+%3C%3D+4

You can see that they fit smoothly together at x=2.