Can an asymptote be an inflection point?

2 Answers
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=(sin(x))/(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:
enter image source here

Mar 29, 2015

The function: #f(x)=1/x# has vertical asymptote: #x=0#

The graph of this function is concave down on #(-oo,0)# and concave up on #(0, oo)#. 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(x)=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.