Understanding Oblique Asymptotes in Mathematical Functions

[flash]

I am trying to draw the graph of

$$y = x + \sqrt{|x|}$$

Can I say that as x approaches infinity, y approaches x? That would mean that the function has an oblique asymptote at the line y=x but I'm not sure.

Thanks for any help!

 

[Redbelly98]

y=x is an asymptote if (and only if):
y-x approaches zero as x approaches infinity.

Try subtracting x from both sides of your equation, and see whether the right-hand-side expression approaches zero or not.

Alternatively, you could plug in successively larger values of x into your expression (i.e, 100, then 1000, then 10,000).
Does the difference between the value and the value of x get larger or smaller?

Eg., for x = 100:
100 + sqrt(100) = ?
This number is ____ larger than 100.
Repeat for 1000, then 10,000.

 

[flash]

Thanks for the reply. Thats what I was thinking, the difference gets larger as x gets larger. But then I thought, when you have a function like x^2 + x, it behaves more and more like x^2 as x gets larger. This function behaves more like x as x gets larger, doesn't it?

 

[Redbelly98]

Both statements are true, these functions "behave more like" x^2 and x, respectively. However, to be an asymptote is a more stringent requirement.

It's likely that, if encountered in a physics or engineering application, you'd be completely justified in approximating the function simply by y=x for large x.