The slope of a tangent line is undefined at points where the function has a vertical tangent. This occurs when the derivative of the function approaches infinity or does not exist. Common examples include points where the function is not continuous or has a cusp or a vertical asymptote.
For example, consider the function f(x) = √x. At the point x = 0, the slope of the tangent line is undefined because the graph has a vertical tangent there. Similarly, functions like f(x) = tan(x) are undefined at points where cos(x) = 0, resulting in vertical asymptotes.
In summary, whenever you encounter a situation where the tangent line turns vertical, the slope is deemed undefined.