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

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Answer 1 · 2018-05-30T15:57:42

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

#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]}