How do you find vertical, horizontal and oblique asymptotes for #[(e^-x)(x^5) + 2] /[ x^5 - x^4 -x +1]#?
Bottom quintic should be factorised in order to find potential vertical asymptotes.
Can be seen easily that 1 and -1 are factors. Then
Comparing coefficients gives,
Another obvious factor of the cubic is -1. Applying the same method,
Then we see,
From this, we can deduce there are horizontal asymptotes as the denominator goes to 0.
Vertical asymptotes at
Oblique and horizontal asymptotes arise as
Because the numerator and denominator are 5th order polynomials,
Due to the nature of exponentials,
This gives us a horizontal asymptote
Using the same limits,
This gives us an oblique asymptote