# How do you find the vertical, horizontal and slant asymptotes of: #y = (2 + x^4)/(x^2 − x^4) #?

Horizontal :

Vertical :

Treating y as a function of x^2, the partial fractions are

directions.

asymptotes.

graph{(2+x^4)/(x^2-x^4) [-40, 40, -20, 20]}

graph{((2+x^4)/(x^2-x^4)-y)(y+1)(x-1-.003y)(x+1+.003y)(x+.001y)=0 [-1.5, 1.5, -15, 15]}

Ad hoc Scale y : x is 20 : 1 for showing asymptotes

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Vertical asymptotes occur where the denominator of the rational function equals zero but the numerator does not. In this case, ( x^2 - x^4 = 0 ), so there are vertical asymptotes at ( x = 0 ) and ( x = \pm 1 ). Horizontal asymptotes can be found by looking at the behavior of the function as ( x ) approaches positive or negative infinity. Since the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote can be found by dividing the leading coefficient of the numerator by the leading coefficient of the denominator. Thus, the horizontal asymptote is ( y = 1 ). There are no slant asymptotes in this case.

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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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