How do you find the asymptotes for #f(x)=(2x)/(sqrt (9x^24))#?
Asymptotes: f(x) = y =
The graph comprises four branches symmetrical about the origin in four regions:
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To find the asymptotes of the function ( f(x) = \frac{2x}{\sqrt{9x^2  4}} ), we need to consider the behavior of the function as ( x ) approaches certain values.

Vertical Asymptotes: Vertical asymptotes occur where the denominator of the function becomes zero, but the numerator does not. In this case, ( 9x^2  4 ) cannot equal zero, so we solve ( 9x^2  4 = 0 ) to find the vertical asymptotes.
[ 9x^2  4 = 0 ] [ 9x^2 = 4 ] [ x^2 = \frac{4}{9} ] [ x = \pm \frac{2}{3} ]
Thus, the vertical asymptotes occur at ( x = \frac{2}{3} ) and ( x = \frac{2}{3} ).

Horizontal Asymptotes: Horizontal asymptotes occur when ( x ) approaches positive or negative infinity. To find them, we examine the behavior of the function as ( x ) approaches infinity and negative infinity.
As ( x ) approaches infinity, the terms with the highest power dominate the function's behavior. In this case, the highest power term is ( x ) in both the numerator and the denominator. Therefore, we can use the ratio of leading coefficients to find the horizontal asymptote.
[ \lim_{x \to \infty} \frac{2x}{\sqrt{9x^2  4}} ]
Since ( x ) is in the numerator and the denominator, the leading coefficients are 2 and 9. Therefore, the horizontal asymptote is ( y = \frac{2}{3} ).
Similarly, as ( x ) approaches negative infinity, the horizontal asymptote is also ( y = \frac{2}{3} ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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