Vertical tangent

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In mathematics, a vertical tangent is intuitively a tangent line with infinite slope, thus being vertical. Generally speaking, vertical tangents occur at vertical asymptotes, but this is not always the case.

[edit] Formal definition

More formally, vertical tangents can be defined using calculus: A function $f (x)$ is said to have a vertical tangent at $x = a$ if and only if $\lim_{x \to a} {f'(x) = \pm \infty}$ .

[edit] Counterexample

An example of a function which has a vertical asymptote but no vertical tangent is

$f(x)=\begin{cases} 1 & x \le 0 \\ \frac{1}{x} & x>0 \end{cases}$

The derivative of this function, then, is

$f'(x)=\begin{cases} 0 & x<0 \\ \mbox{undefined} & x=0 \\ -\frac{1}{x^2} & x>0 \end{cases}$

In this function, there is a vertical asymptote at $x = 0$ because $\lim_{x \to 0^+} {f'(x)=-\infty}$ . There is no vertical tangent here because the undirected limit of $f^{\prime}(x)$ does not exist as $x \to 0$ .