Definition:Vertical Tangent Line

Definition
Let $P = \tuple {c, \map f c}$ be a point on the graph of a real function $f$.

The vertical line $x = c$ is a vertical tangent line to the graph of $f$ at $P$ any of the following hold:


 * $(1): \quad f$ is right continuous at $c$ and $\ds \lim_{x \mathop \to c^+} \map {f'} x = +\infty$


 * $(2): \quad f$ is right continuous at $c$ and $\ds \lim_{x \mathop \to c^+} \map {f'} x = -\infty$


 * $(3): \quad f$ is left continuous at $c$ and $\ds \lim_{x \mathop \to c^-} \map {f'} x = +\infty$


 * $(4): \quad f$ is left continuous at $c$ and $\ds \lim_{x \mathop \to c^-} \map {f'} x = -\infty$