Why can horizontal asymptotes be crossed?

  1. 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!
     
  2. jcsd
  3. 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.
     
  4. HallsofIvy

    HallsofIvy 40,556
    Staff Emeritus
    Science Advisor

    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
    [tex]f(x)= xe^{-x^2}[/tex]
    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.
     
    Last edited: Aug 5, 2007
  5. Horizontal Asymptotes only describe end behavior, so as long as the graph tends towards the value eventually, its alright if its crossed.
     
  6. 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.
     
  7. I don't think I would exactly call that crossing.
     
  8. The simplest example I can think of is [tex] \frac{x}{{x^{2}} + 1} [/tex].
    y = 0 is a horizontal asymptote, however, the function is equal 0 at x = 0.
     
  9. HallsofIvy

    HallsofIvy 40,556
    Staff Emeritus
    Science Advisor

    Another, more complicated, example is
    [itex]f(x)= xe^{-x^2}[/itex]
     
  10. As far as rational functions go, the function can only switch directions a limited number of times. (This is based on the number of places the derivative is zero, and the number of vertical asymptotes.) As we look at the function going in the x direction, the function can cross its horizontal asymptote as long as it can turn back around and tend towards it at infinity. To put it another way, the function can cross this horizontal asymptote as long as you are not beyond all of the possible turning points. Beyond the turning points, the function can no longer cross the asymptote.
     
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