Defining vertical tangent lines

Question

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

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

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

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

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 $a$.

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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 Educators Stack Exchange.)

Answer 1

Speaking as a geometer, I want "tangency" to be independent of the coordinate system. Particularly, if $f$ is a real-valued function of one variable defined in some neighborhood of $a$, and if $f$ is invertible in some neighborhood of $a$, then the line $x = a$ should be tangent to the graph $y = f(x)$ at $a$ if and only if the line $y = b = f(a)$ is tangent to the graph $y = f^{-1}(x)$ at $b$.

For an elementary calculus course I'd want:

• $f$ continuous in some neighborhood of $a$;

• $f$ invertible in some neighborhood of $a$;

• $f'(a) = \pm\infty$, i.e., $(f^{-1})'(b) = 0$ (the graph $y = f^{-1}(x)$ has $y = a$ as horizontal tangent).

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Condition 1 does not guarantee invertibility near $a$ (as the cusp shows), so in my book it's out.

Condition 2 implies all three items of my wish list. ($f$ is implicitly assumed differentiable in some neighborhood of $a$; the derivative condition guarantees the derivative doesn't change sign in some neighborhood of $a$, and that $f'(a) = \pm\infty$.)

Condition 3 does not imply continuity (as the step function shows), so it's out.

Answer 2

This isn't a completely accurate answer, but it works for enough of the time that I'll post it anyway.

You know that in normal circumstances of $y$ as a function of $x$ where you want to find a HORIZONTAL tangent line, one way is to find

${dy \over dx} = 0$

and solve for values of $x$ for which this is the case (note: the function and its derivative must be continuous at the points where the derivative equals 0 to warrant a horizontal tangent.)

In reality, the equation above is asking "at what points $x$ does $y$ not change when $x$ changes?"

But if I'm looking at a VERTICAL tangent line, I'm really asking the equation "at what points $x$ does $x$ not change when $y$ changes?"

So then instead of $dy \over dx$, I want to know instead when

${dx \over dy} = 0$

for which both $x$ and its derivative are continuous at the points where ${dx \over dy} = 0$.

Vertical tangents are more difficult to define when we take all y as a function of x because (unless the function is one-to-one) it is difficult to define a vertical cusp. Instead, if we define a vertical tangent line to exist when:

• $x$ is continuous and
• its derivative $dx \over dy$ is continuous

we can rule out vertical cusps, which do not have a continuous derivative.

While I don't know of a good standard definition, I hope this one I was taught helps you understand what vertical tangents are.