Not all functions have Asymptotes, right?

[Sabellic]

I was just wondering about asymptotes. I know that the symptotes for 1/f(x) for example is established by isolating f(x) and setting f(x)=0.

However, I was wondering if non-reciprocal functions such as f(x)=x or f(x)=x^2 have asymptotes. I was thinking "no" because there is no axis that they approach but do not intersect.

 

[Defennder]

ln x which is a non-reciprocal function has a vertical aymptote at x=0. And so do tan x, sec x, cosec x.

EDIT: Okay, so maybe tan x, sec x and cosec x doesn't count because they are defined as a fraction of functions.

The two functions you describe do not have asymptotes.

 

[Dick]

f(x) can be said to have an asymptote, y=x, if you are generous in the wording of the definition of asymptote and allow asymptotes that aren't vertical or horizontal. f(x)=x^2 definitely doesn't have any. exp(x) has a horizontal asymptote. Isn't that a 'non-reciprocal' function? You just need to be clear about the exact definition of asymptote and apply it to each function.

 

[Sabellic]

Oh, that's right. f(x)=2^x does have a horizontal asymptote. And yes, I was thinking more of having either horizontal or vertical asymptotes.

Thank you very much.

How about the case of f(x)=square root of x, or f(x)=absolute value of x?

Because these the former touches the y-axis but can't have a x value less than zero; whilst the latter touches the x-axis but can't have a y value less than zero. Can these be defined as asymptotes or would I just define their limits in the set notation?

 

[Dick]

f(x)=sqrt(x), no. For f(x)=|x|, again, you have to look at the exact wording of your definition of 'asymptote'. It's more of a technical legal question. Not everyone uses exactly the same one all the time.