Is $x^2$ function differentiable at $x=0$?

Question

My book says that $x^2$ function is differentiable at $x=0$. How is this possible, given that the right limit is greater than zero and the left limit is less than zero?

Answer 1

Both limits are $0.$ The function $x^2$ is always positive regardless of the sign of $x.$ Thus, it has to be the case that the right limit must be equal to the left limit as well.

Answer 2

See them graphically....

The absolute value has a corner at the origin, it changes direction suddently

The parabola $x^2$ changes direction gradually