# How do you find the asymptotes for #y=(2x^2+3)/(x^2-6)#?

vertical asymptotes at

horizontal asymptote at y = 2

The denominator of y cannot be zero as this is undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.

Horizontal asymptotes occur as

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To find the asymptotes for the function ( y = \frac{2x^2 + 3}{x^2 - 6} ), we examine the behavior of the function as ( x ) approaches positive or negative infinity.

- Vertical asymptotes occur where the denominator of the rational function becomes zero, but the numerator does not. Therefore, set the denominator equal to zero and solve for ( x ).

[ x^2 - 6 = 0 ]

Solving for ( x ):

[ x^2 = 6 ]

[ x = \pm \sqrt{6} ]

So, there are vertical asymptotes at ( x = \sqrt{6} ) and ( x = -\sqrt{6} ).

- Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator. In this case, the degrees of the numerator and denominator are both 2. To find the horizontal asymptote(s), divide the leading term of the numerator by the leading term of the denominator.

The leading term of the numerator is ( 2x^2 ) and the leading term of the denominator is ( x^2 ).

Thus, the horizontal asymptote is ( y = \frac{2}{1} = 2 ).

Therefore, the function has two vertical asymptotes at ( x = \sqrt{6} ) and ( x = -\sqrt{6} ), and one horizontal asymptote at ( y = 2 ).

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To find the asymptotes of the function ( y = \frac{2x^2 + 3}{x^2 - 6} ):

- Determine the degree of the numerator and the denominator.
- If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at ( y = 0 ).
- If the degree of the numerator is equal to the degree of the denominator, divide the coefficients of the highest degree terms to find the horizontal asymptote.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, perform polynomial long division to find any slant asymptotes.
- Set the denominator equal to zero and solve for ( x ) to find vertical asymptotes.

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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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