Definition:Vertical Tangent Line

Definition
Let $P = \left({c, f \left({c}\right)}\right)$ 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 $\displaystyle \lim_{x \mathop \to c ^+} f' \left({x}\right) = +\infty$


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


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


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