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

There is no vertical asymptote, as the denominator will always be unequal to

In "the language":

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To find the asymptotes of the function ( f(x) = \frac{x}{x^2 + 4} ), we need to determine both the vertical and horizontal asymptotes.

Vertical Asymptotes: Vertical asymptotes occur where the denominator of the function becomes zero but the numerator does not. Set the denominator equal to zero and solve for x:

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

Since no real number squared can equal a negative number, this equation has no real solutions. Therefore, there are no vertical asymptotes for the function.

Horizontal Asymptotes: To find horizontal asymptotes, we look at the behavior of the function as x approaches positive or negative infinity. We compare the degrees of the numerator and denominator:

Degree of numerator: 1 (x) Degree of denominator: 2 (x^2)

Since the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y = 0 (the x-axis).

Therefore, for the function ( f(x) = \frac{x}{x^2 + 4} ), there are no vertical asymptotes, and the horizontal asymptote is 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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