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In looking at the definition of vertical tangent lines in some popular calculus texts,

I noticed that there are a few different definitions for this term, including the following:

A function $f$ has a vertical tangent line at $a$ if

  1. $\;f$ is continuous at $a$ and $\displaystyle\lim_{x\to a}\;\lvert f^{\prime}(x)\rvert=\infty$

  2. $\;f$ is continuous at $a$ and $\displaystyle\lim_{x\to a} f^{\prime}(x)=\infty$ or $\displaystyle\lim_{x\to a} f^{\prime}(x)=-\infty$

  3. $\;\displaystyle\lim_{h\to0}\frac{f(a+h)-f(a)}{h}=\pm\infty$

I would like to ask if there is a standard definition of this term, and whether or not the definition should include continuity at $a$ and should not include the situation where the graph has a vertical cusp at $a$.


Here are some examples where these definitions lead to different conclusions:

a) $\;f(x)=x^{2/3}$

b) $\;f(x)=\begin{cases}1&\mbox{, if }x>0\\0&\mbox{, if }x=0\\-1&\mbox{, if }x<0\end{cases}$

(This question has also been posted on Math Stack Exchange.)

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  • $\begingroup$ It took me time to understand that you mean $x\mapsto x^{\frac23}$ to be defined over $\mathbb{R}$; I do not know if this is customary in some countries, but I believe it a bad idea to define any non-integer power of negative numbers. It yields very different points of view for $x^{\frac12}$, $x^{\frac17}$ and $x^{\sqrt{2}}$ which can only add to the usual confusions on powers. $\endgroup$ – Benoît Kloeckner Jan 25 '16 at 7:55
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    $\begingroup$ A simple approach would be to require that the function be invertible in some neighborhood of $a$, and that the inverse function have a zero derivative. Another method would be to use a parametrized curve, and define the tangent line as the one that points in the direction of $(\dot{x},\dot{y})$. $\endgroup$ – Ben Crowell Jan 25 '16 at 16:45
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    $\begingroup$ Related: math.stackexchange.com/questions/771614/… $\endgroup$ – Ben Crowell Jan 25 '16 at 16:47
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I suppose if one had a definition of a tangent line, defined for all curves, then one would only need to know which way is vertical. [Thurston, On the Definition of a Tangent-Line (1964)] explores several definitions. A geometric one is the following, where an arc is defined to be the homeomorphic image of a closed, bounded interval.

The line $L$ through a point $C$ of an arc is a geometrical tangent to the arc at $C$ if, given any positive number, there is a circle with centre $C$ such that the angle between the line $PC$ and the line $L$ is less than the given number for every point $P$ (other than $C$) which is both on the arc and in the circle.

This would mean there is a tangent line at a cusp, and, in the case of $y = f(x)$, it is equivalent to def. 1. Thurston points out, more or less, that in the parametric approach one requires that "the ratio $X'(c) \colon Y'(c)$ exists," which it won't at a cusp.

The OP asks about tangent line explicitly in terms of a function and not of curve. It is easy to talk about a nonvertical tangent in terms of the explicit linear function $f(c) + f'(c)\,(x-c)$, provided $f'(c)$ is defined. Thurston calls the corresponding line an "explicit tangent." But it seems to hard me to talk about vertical tangents without considering the graph of the function as a curve in the plane, except strictly in terms a definition like one those given in the OP. Note, too, that the definition of the tangent line as the limit of the secant line leads to def. 1, assuming that the notion of the limiting position of a variable line can be made sufficiently clear.

I guess I have tended to use def. 1 myself, probably because of the textbooks I've used. It's easier for me to adapt to the textbook than for the students to adapt to my idiosyncrasies, especially first-year students. But I could live with def. 2. I don't believe there is consensus about a standard definition. It's not clear to me how much the parametric situation should influence the single-variable function case. It's fairly clear to me that the distinction between def. 1 and 2 in first-year calculus is not that important (except in each particular class, terminology and instructions have to be clear).

An interesting aside, not particularly relevant to the OP's question, is that while any explicit tangent line $y = f(c) + f'(c)\,(x-c)$ is equivalent to a tangent line according to the parametric definition, there is a parametrization of a curve that has a tangent line at a certain point $C$ but which is not locally equivalent to a graph $y = f(x)$ with an explicit tangent at $C$. The parametrization involves one of my favorite functions and I leave the reader to ponder it (or read Thurston's paper):

$$ \cases{X(t) = t^2 \sin t^{-1} + {1\over2}\,t & if $0<|t|\le1$, and $X(0)=0$ \cr Y(t) = t\,X(t)\cr} $$

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Personally, I like option $2$.

As you point out, $1$ is satisfied by functions with cusps as in your example $a$, which does not match up with my intuition about tangent lines.

$3$ is the wrongest, because of your example $b$.

I do not think the definition is standardized however. As is often the case in mathematics, it is probably best just to clarify what you mean if one of these situations arise. Definitions change from person to person.

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    $\begingroup$ It'd be nice if the definition of horizontal and vertical tangents were equivalent under an exchange of $x$ and $y$. From a parametric perspective they ought to be equivalent. $\endgroup$ – James S. Cook Jan 21 '16 at 3:02
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    $\begingroup$ @JamesS.Cook I think that if $f$ satisfies $2$, then it is locally invertible and its inverse has a vertical tangent. Of course, you are probably correct that these concepts really apply more to graphs than to functions, and the function case should be a special case of more general curves. So perhaps the best definition is ``A curve $C$ has a vertical tangent at $p$ if there is a local parameterization $\gamma: [-1,1] \to C$ with $\gamma(0)=p$, $\gamma$ differentiable at $p$, and $\gamma'(p) = (0,1)$'' (one could always renormalize to $(0,1)$ if it was $(0,y)$ with $y \neq 0$). $\endgroup$ – Steven Gubkin Jan 21 '16 at 3:45
  • $\begingroup$ I am relatively in agreement with this answer, without wanting to dismiss definition 1 entirely; it could be said that a function satisfying 1 has a vertical tangent half-line. $\endgroup$ – Benoît Kloeckner Jan 25 '16 at 7:57
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Suppose function $f$ is continuous at $a$. Then $f$ has a vertical tangent line at $a$ iff $$\displaystyle\lim_{h\to0}\frac{h}{f(a+h)-f(a)}=0$$

i.e. the inverse of the slope of the tangent there will be $0$.

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