# Can an even function have an oblique asymptote? If yes, provide example/graph of a function with an oblique asymptote. If not, explain why.

degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted on

I would also use quantatanu's example of `f(x)=|x|`, and here's the graph.

The oblique asymptotes are the lines `y=x`(as `x->oo`) and `y=-x` (as `x->-oo` ) If this seems like a strange example, it's probably because there are some common misconceptions about asymptotes. It's not true that an asymptote is a line that the function "gets closer to but never touches."

As long as the distance between the graph of the function and a line approaches zero as x approaches plus or minus infinity, that line is an asymptote. Here, the distance between the graph of `|x|` and `x` is always zero if `x>0` , so it is certainly an asymptote. Likewise for the other asymptote in the negative direction.

Maybe you already knew that, but I figured I should add something of my own and not just copy the previous answer.

Sources:

degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted on

In that case, the example can easily be modified to something like `y=|x+1/x|,` which still has the asymptotes `y=x` and `y=-x.`

There are infinitely many others as well. If the graph of some function `f` has the oblique asymptote `L` (say as `x->+oo)` , then
define `g(x)=f(|x|).` The new function `g` will be even and have the same slant asymptote (and the reflection of `L` across the `y` axis is another slant asymptote).

aasra | Student, Undergraduate | (Level 1) eNoter

Posted on

My teacher has taught us that an asymtote is indeed a line that the function approaches but never touches, and that is why it would get difficult to go with the answers given by quantatanu and degeneratecircle, and try to prove myself.

quantatanu | Student, Undergraduate | (Level 1) Valedictorian

Posted on

The simplest example is

f(x) = |x|

which is an even function and its assymptotes are oblique. I am not an editor so unable to paste the graph here, but let me try just by using text, to plot |x|

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Fig.1. Plot of |x|

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Fig.2. Assymptotes of |x|