# Why can horizontal asymptotes be crossed?

I thought that horizontal asymptotes were asymptotes and now I'm hearing that they can be crossed... Is this true? If so, why and how?

Okay, wait a second, I mean horizontal asymptotes. According to: http://www.purplemath.com/modules/asymtote2.htm

"As I mentioned before, it is common and perfectly okay to cross a horizontal asymptote. It's the verticals that you're not allowed to touch." In what cases are horizontal asymptotes crossed? I've never encountered such a thing before... at least I think. Please give me a simple equation or something. Thanks in advance!

f(x)=sin(x)/x

As x approaches infinity, f(x) obviously approaches zero, however, as x gets larger you can always find points where f(x) is positive(let x=(4n+1)pi/2) and other points where f(x) is negative(let x=(4n+3)pi/2). So we have a function that approaches the horizontal assymptote y=0, yet crosses that assymptote an infinite number of times.

HallsofIvy
Homework Helper
Look at the definition of "horizontal asymptote"- a horizontal line that the function gets closer and closer to as x goes to plus or minus infinity. That says nothing about what happens for finite values of x. The reason that can't happen with vertical asymptotes is that a function can have only one value for a give x but can can have many x values that give the same y.

An example is
$$f(x)= xe^{-x^2}$$
The graph crosses the x axis at x=0. For x> 0, it rises to a maximum value and then decreases toward y= 0 as x goes to infinity. For x< 0, it decreases to a minimum values then rises toward y= 0 as x goes to negative infinity. y= 0 is a horizontal asymptote but the graph crosses y= 0 at x= 0. Notice that the function takes on any value of y between the minimum and maximum values twice.

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Horizontal Asymptotes only describe end behavior, so as long as the graph tends towards the value eventually, its alright if its crossed.

A function can cross its vertical asymptote, though not more than once and certainly not infinitely many times like it can its horizontal asymptote. For example, f(x) := 1/x for x !=0 and f(0) := 0.

A function can cross its vertical asymptote, though not more than once and certainly not infinitely many times like it can its horizontal asymptote. For example, f(x) := 1/x for x !=0 and f(0) := 0.
I don't think I would exactly call that crossing.

The simplest example I can think of is $$\frac{x}{{x^{2}} + 1}$$.
y = 0 is a horizontal asymptote, however, the function is equal 0 at x = 0.

HallsofIvy
$f(x)= xe^{-x^2}$