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

Horizontal :

Vertical :

There is no quotient in this division. So, existence of slant asymptote

is ruled out..

Cross multiplying,

If one factor in LHS is 0, the other has to be infinite, when the

Now, y = 0 gives horizontal asymptote.

The other factor has a factor (x +1 ) and its other factor has complex

zeros.

So, x + 2 = 0 gives the vertical asymptote.

graph{ y(y (3x^3-2x+1) - 2x^2+1)=0 [-8, 8 -3.26, 3.276]} #

See the Socratic graph.

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To find the vertical asymptotes of a rational function, you need to identify the values of ( x ) that make the denominator equal to zero, but the numerator doesn't. For the given function ( \frac{2x^2 - 1}{3x^3 - 2x + 1} ), you can solve ( 3x^3 - 2x + 1 = 0 ) to find potential vertical asymptotes. However, this cubic equation might not have real solutions, and there may not be any vertical asymptotes.

Horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ( y = 0 ). If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. In this case, the degree of the numerator is 2, and the degree of the denominator is 3, so there is no horizontal asymptote.

To find slant asymptotes, you perform polynomial long division to divide the numerator by the denominator. If the degree of the resulting quotient is one less than the degree of the denominator, then there is a slant asymptote. If not, there is no 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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