Question

What are the asymptote(s) and hole(s), if any, of `f(x) =(sinx+cosx)/(x^3-2x^2+x)`?

Answer 1

`x=0` and `x=1` are the asymptotes. The graph has no holes.

Explanation:

`f(x) =(sinx+cosx)/(x^3-2x^2+x)`

Factor the denominator:

`f(x) =(sinx+cosx)/(x(x^2-2x+1))`

`f(x) =(sinx+cosx)/(x(x-1)(x-1))`

Since none of the factors can cancel out there are no "holes", set the denominator equal to 0 to solve for the asymptotes:

`x(x-1)(x-1)=0`

`x=0` and `x=1` are the asymptotes.

graph{(sinx+cosx)/(x^3-2x^2+x) [-19.5, 20.5, -2.48, 17.52]}