# How do you find the asymptotes for #y = (3x^2+x-4) / (2x^2-5x) #?

The vertical asymptotes are

No hole, No slant asymptote

The horizontal asymptote is

Let's factorise the denominator

For the limits, we take the terms of highest degree

graph{(y-((3x^2+x-4)/(2x^2-5x)))(y-3/2)=0 [-14.24, 14.24, -7.12, 7.12]}

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To find the asymptotes of the rational function ( y = \frac{3x^2 + x - 4}{2x^2 - 5x} ), we need to examine the behavior of the function as ( x ) approaches positive and negative infinity.

**Vertical Asymptotes**: Vertical asymptotes occur where the denominator of the function becomes zero, but the numerator does not. To find the vertical asymptotes, set the denominator equal to zero and solve for ( x ).

[ 2x^2 - 5x = 0 ]

Factor out ( x ):

[ x(2x - 5) = 0 ]

So, ( x = 0 ) or ( 2x - 5 = 0 ):

[ 2x - 5 = 0 ] [ 2x = 5 ] [ x = \frac{5}{2} ]

Therefore, the vertical asymptote is ( x = 0 ) and ( x = \frac{5}{2} ).

**Horizontal Asymptotes**: Horizontal asymptotes occur when the degree of the numerator is equal to or less than the degree of the denominator. To find horizontal asymptotes, compare the degrees of the numerator and denominator.

The degree of the numerator is 2 and the degree of the denominator is also 2.

To find the horizontal asymptote(s), divide the leading term of the numerator by the leading term of the denominator:

[ \frac{3x^2}{2x^2} = \frac{3}{2} ]

So, the horizontal asymptote is ( y = \frac{3}{2} ).

Therefore, the asymptotes for the function ( y = \frac{3x^2 + x - 4}{2x^2 - 5x} ) are:

- Vertical asymptotes: ( x = 0 ) and ( x = \frac{5}{2} )
- Horizontal asymptote: ( y = \frac{3}{2} )

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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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