Question

What are the asymptotes of the function `f(x) = x+1+sin(x^2)/x` ?

Answer 1

See explanation...

Explanation:

Note that for any real value of `x` we have `abs(sin(x^2)) <= 1`

So:

`abs(sin(x^2)/x) <= abs(1/x) -> 0` as `x->oo`

So the function:

`f(x) = x+1+sin(x^2)/x`

tends to the line `x+1` as `x->oo`.

However, note that whenever `x = sqrt(npi)` for some integer `n`, then `sin(x^2) = 0`.

Hence the function `f(x) = x+1+sin(x^2)/x` and the line `x+1` intersect at infinitely many separated points.

Some definitions of asymptote preclude the intersection at infinitely many points, but most modern definitions allow it.