# How do you find the vertical, horizontal or slant asymptotes for #(x^3-x+1)/(2x^4+x^3-x^2-1)#?

This function has horizontal asymptote

It has no slant asymptotes or holes.

Given:

These two real zeros are potentially vertical asymptotes. They can only fail to be so if the numerator is zero at them (which could result in a hole instead). This happens if the numerator and denominator have a common factor - so let's find their GCF using Euclid's method...

Since we found no exact division, the GCF is constant. There is no common linear or higher degree factor.

Slant (a.k.a. oblique) asymptotes of a rational function can only arise if the degree of the numerator is one more than that of the denominator. Such is not the case in our example.

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To find the vertical asymptotes of a rational function, we need to determine the values of ( x ) for which the denominator equals zero. These values will give us the ( x )-coordinates of the vertical asymptotes. In this case, we set the denominator ( 2x^4 + x^3 - x^2 - 1 ) equal to zero and solve for ( x ).

To find horizontal or slant asymptotes, we need to compare 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 the ( x )-axis (y = 0). If the degree of the numerator is equal to the degree of the denominator, we can find the horizontal asymptote by dividing the leading coefficients. If the degree of the numerator is greater than the degree of the denominator by exactly one, there is a slant asymptote. To find it, perform polynomial long division.

To summarize:

- Vertical asymptotes: Set the denominator equal to zero and solve for ( x ).
- Horizontal asymptote: Compare the degrees of the numerator and denominator.
- Slant asymptote: If the degree of the numerator is greater than the degree of the denominator by exactly one, perform polynomial long division to find the slant 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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