# How do you find the asymptotes for #F(x) = (-2x^2 + 1)/(2x^3 + 4x^2)#?

Vertical:

For visual effect, see the graph.

to two ( of the three ) branches of the graph, in the two opposite directions.

graph{2yx^2(x+2)+2x^2-1=0 [-10, 10, -5, 5]}

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To find the asymptotes for the function ( F(x) = \frac{-2x^2 + 1}{2x^3 + 4x^2} ), follow these steps:

- Identify vertical asymptotes by setting the denominator equal to zero and solving for ( x ). These values are excluded from the domain due to division by zero.

[ 2x^3 + 4x^2 = 0 ]

[ 2x^2(x + 2) = 0 ]

[ x = 0 ] (double root)

- Identify horizontal or slant asymptotes. For rational functions, 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, there is no horizontal or slant asymptote.

In this case, the degree of the numerator is less than the degree of the denominator, so there is a horizontal asymptote at ( y = 0 ).

Therefore, the vertical asymptote is at ( x = 0 ) and the horizontal asymptote is at ( y = 0 ).

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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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