# Can an asymptote be an inflection point?

Mar 29, 2015

Since an inflection point is a point on an equation, I assume you mean
"Can an asymptote intersect the line of an equation at an inflection point?"

Under current/modern usage of the concept of asymptote, the answer is a simple "yes";
for example, $y = 0$ is considered an asymptote of $y = \frac{\sin \left(x\right)}{{x}^{2}}$

Older, more traditional definitions of "asymptote" included a restriction that the equation could not cross the asymptote infinitely; so the given example would not be valid.

However it is possible to imagine a situation like that pictured below
which would still be valid under traditional definitions:

Mar 29, 2015

The function: $f \left(x\right) = \frac{1}{x}$ has vertical asymptote: $x = 0$

The graph of this function is concave down on $\left(- \infty , 0\right)$ and concave up on $\left(0 , \infty\right)$. The concavity changes at the asymptote.

I have known students to incorrectly say that $x = 0$ is an inflection point. If that is the intended question, then:

Note that: an inflection point is a point on the graph where the concavity changes. There is no point of the graph of $f \left(x\right) = \frac{1}{x}$ at which the concavity changes, so the graph has no inflection point.

As Alan P. said in his answer, a graph can have a point of inflection that lies on its asymptote.