[Math] Defining vertical tangent lines

calculusdefinitionsoft-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 $a$.

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.)

Best Answer

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).

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.

Related Question