# How do you find the Vertical, Horizontal, and Oblique Asymptote given #f(x)=(6x^2+2x-1 )/ (x^2-1)#?

The vertical asymptotes are

The horizontal asymptote is

No oblique asymptote

For the limit, we take the term with the highest coefficient

graph{(y-(6x^2+2x-1)/(x^2-1))(y-6)=0 [-12.83, 12.49, -1.84, 10.83]}

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To find the vertical asymptotes of a rational function ( f(x) = \frac{{6x^2 + 2x - 1}}{{x^2 - 1}} ):

- Set the denominator equal to zero and solve for ( x ).
- Any value of ( x ) that makes the denominator zero is a vertical asymptote.

To find the horizontal asymptotes:

- Compare the degrees of the numerator and denominator.
- If the degree of the numerator is greater, there is no horizontal asymptote.
- If the degree of the denominator is greater, the horizontal asymptote is ( y = 0 ).
- If the degrees are equal, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote.

To find the oblique asymptote:

- If the degree of the numerator is exactly one more than the degree of the denominator, there is an oblique (slant) asymptote.
- Use polynomial long division or synthetic division to divide the numerator by the denominator.
- The quotient obtained is the equation of 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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- How do you find the asymptotes for #(2x^2 - x - 38) / (x^2 - 4)#?
- How do I find the domain of #y=4-x^2#?

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