# How do you find vertical, horizontal and oblique asymptotes for #[(e^-x)(x^5) + 2] /[ x^5 - x^4 -x +1]#?

Define

Bottom quintic should be factorised in order to find potential vertical asymptotes.

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.

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To find the vertical asymptotes, determine the values of ( x ) that make the denominator zero, excluding any values that make the numerator zero as well. For horizontal asymptotes, look at the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ( y = 0 ). If the degrees are equal, divide the leading coefficients to find the horizontal asymptote. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. To find oblique asymptotes, perform polynomial long division to divide the numerator by the denominator and determine the quotient. The quotient represents the oblique asymptote.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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