Getting Closer And Closer

Algebra Level 1

True or False?

$\quad$ A graph of a function can cross an asymptote .

True False

3 solutions

Kai Ott
Aug 8, 2016

Relevant wiki: Asymptotes

Let $f(x)= a(x) + \frac {p(x)}{q(x)}$ where $a, p, q$ are polynomial functions and where the degree of $q$ is greater than the degree of $p$ . In this example $a(x)$ is the asymptote and $f(x)$ crosses it (which means $f(x) = a(x)$ )for all $p(x)=0$ .

An asymptote of a curve is a line to which the curve converges. In other words, the curve and its asymptote get infinitely close, but they never meet.

Above is the definition of asymptote in wiki. Wouldn't your claim violate the definition? If no, please explain with a proper example.

Mayank Jain - 4 years, 10 months ago

Well the function might intersect with the asymptote before convergence. And the convergence happens later.

Adrian Pen - 4 years, 10 months ago

As an explicit example take $f(x)=1+ \frac{x-1}{x^2}$ . As x tends to infinity the graph of the function converges to the asymptote y=1. Note that the function crosses that asymptote at x=1.

Kai Ott - 4 years, 10 months ago

Joe Potillor
Aug 22, 2016

Functions cam of course go through horizontal and oblique aysmptotes, but not vertical aysmptotes

Paulo Correia
Oct 28, 2016

$f(x)=\dfrac{\sin x}{x}$

https://www.wolframalpha.com/input/?i=assomptote+(sin+x)%2Fx

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