# How do you find the asymptotes for #f(x) = (3x^2 + 15x +18) /( 4x^2-4)#?

The vertical asymptotes are

No slant asymptote

The horizontal asymptote is

We need

We factorise the denominator

graph{(y-(3x^2+15x+18)/(4x^2-4))(y-3/4)=0 [-14.24, 14.23, -7.12, 7.11]}

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To find the asymptotes of ( f(x) = \frac{3x^2 + 15x + 18}{4x^2 - 4} ), you need to identify the vertical and horizontal asymptotes:

Vertical asymptotes occur where the denominator equals zero and the numerator doesn't. To find them, solve (4x^2 - 4 = 0) for (x).

(4x^2 - 4 = 0) simplifies to (x^2 - 1 = 0). Solve this quadratic equation to find (x).

(x^2 - 1 = 0) factors to ((x - 1)(x + 1) = 0).

So, (x = 1) and (x = -1) are the vertical asymptotes.

Horizontal asymptotes occur when the degree of the numerator is the same as the degree of the denominator. In this case, both numerator and denominator have the same degree (2).

So, divide the leading coefficients of both numerator and denominator.

The horizontal asymptote is (y = \frac{3}{4}).

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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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