Definition:Vertical Tangent Line

Definition

Let $P = (c, f \left({c}\right))$ 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 $P$ iff any of the following hold:

• $f$ is right continuous at $c$ and $\displaystyle \lim_{x \to c ^+} f^{\prime} \left({x}\right) = +\infty$

• $f$ is right continuous at $c$ and $\displaystyle \lim_{x \to c ^+} f^{\prime} \left({x}\right) = -\infty$

• $f$ is left continuous at $c$ and $\displaystyle \lim_{x \to c ^-} f^{\prime} \left({x}\right) = +\infty$

• $f$ is left continuous at $c$ and $\displaystyle \lim_{x \to c ^-} f^{\prime} \left({x}\right) = -\infty$

Sources

• Roland E. Larson,  Robert P. Hostetler and Bruce H. Edwards: Calculus: 8th Edition (2005): $\S 2.1$

• Weisstein, Eric W. "Vertical Tangent." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/VerticalTangent.html