It's well known that the derivative of the function $|x|$ is not defined at $x = 0$ because the left and right limits are different. But consider that the tangent line is a line that touches a curve in only one point. Even though the derivative doesn't exist, $y = 0$ meets the requirements of being a tangent line of $|x|$ at $x = 0$, doesn't it? What am i missing on the definition of a derivative as the slope of the tangent line? Is there an extra condition not being considered?
What is the exact definition of the tangent line of a curve in the derivative context?