How do you find vertical, horizontal and oblique asymptotes for #(3x^2+2x-1 )/(x^2-4)#?
Oblique asymptotes occur when the degree of the numerator is greater than the degree of the denominator. This is not the case here hence no oblique asymptote. graph{(3x^2+2x-1)/(x^2-4) [-10, 10, -5, 5]}
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To find the vertical, horizontal, and oblique asymptotes for the function ( \frac{3x^2+2x-1}{x^2-4} ):
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Vertical Asymptotes: Vertical asymptotes occur where the denominator of the function is equal to zero, but the numerator is not. In this case, the denominator is (x^2 - 4), which factors to ((x-2)(x+2)). Therefore, the vertical asymptotes are (x = 2) and (x = -2).
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Horizontal Asymptotes: 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 are equal (both 2), so we need to compare the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1. Therefore, there is a horizontal asymptote at (y = \frac{3}{1} = 3).
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Oblique Asymptotes: Oblique (or slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. In this case, the degree of the numerator is 2 and the degree of the denominator is 2. Therefore, there are no oblique asymptotes for this function.
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To find vertical asymptotes, set the denominator equal to zero and solve for x. In this case, the denominator is (x^2 - 4), so setting it equal to zero gives us (x = 2) and (x = -2). These are the vertical asymptotes.
Horizontal asymptotes are found by examining the behavior of the function as x approaches positive or negative infinity. For rational functions where the degree of the numerator is equal to or less than the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. In this case, the horizontal asymptote is (y = 3).
Oblique asymptotes occur when the degree of the numerator is one greater than the degree of the denominator. To find the oblique asymptote, perform polynomial long division or synthetic division to divide the numerator by the denominator. The quotient obtained will represent the oblique asymptote.
So, to summarize:
- Vertical asymptotes: (x = 2) and (x = -2)
- Horizontal asymptote: (y = 3)
- Oblique asymptote: Perform polynomial long division or synthetic division to find it.
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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.
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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