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 $\ds \lim_{x \mathop \to c^-} \map {f'} x = -\infty$


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