Discontinuities of The Second Kind - The Vertical Tangent

The vertical tangent is a type of discontinuity that depends on not the behavior of the actual function, but its derivative.

We know that the derivative is defined as the slope of the line tangent to the graph of the function at a given point. Essentially, the instantaneous rate of change of that function at that point.

Let's consider an example:
Take the derivative.
Take its limit to x = 0.

The limit of the derivative from both sides exists and is an infinite. This is the definition of a vertical asymptote.