Vertical tangent

Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency.

A function ƒ has a vertical tangent at x = a if the difference quotient used to define the derivative has infinite limit: The graph of ƒ has a vertical tangent at x = a if the derivative of ƒ at a is either positive or negative infinity.

For a continuous function, it is often possible to detect a vertical tangent by taking the limit of the derivative.

Similarly, if then ƒ must have a downward-sloping vertical tangent at x = a.

This occurs when the one-sided derivatives are both infinite, but one is positive and the other is negative.