Definition:Vertical Tangent Line
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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 $P$ if and only if 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 $\ds \lim_{x \mathop \to c^-} \map {f'} x = -\infty$
Sources
- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 2.1$
- Weisstein, Eric W. "Vertical Tangent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VerticalTangent.html