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

May 30, 2018

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

#### Explanation:

$f \left(x\right) = \frac{\sin x + \cos x}{{x}^{3} - 2 {x}^{2} + x}$

Factor the denominator:

$f \left(x\right) = \frac{\sin x + \cos x}{x \left({x}^{2} - 2 x + 1\right)}$

$f \left(x\right) = \frac{\sin x + \cos x}{x \left(x - 1\right) \left(x - 1\right)}$

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

$x \left(x - 1\right) \left(x - 1\right) = 0$

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

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