How do you find vertical, horizontal and oblique asymptotes for #[(3x^2) + 14x + 4] / [x+2]#?
Vertical:
Oblique:
Reorganizing to the form (ax+bx+c((a'x+b/y+c')=k,
This represents a hyperbola with asymptotes
The first graph is asymptotes inclusive and the second is for the
hyperbola, sans asynptotes.
Note: The second degree equation
is so for the given equation.
graph{(y3x8)(x+2)((y3x8)(x+2)+14)=02 [80, 80, 40, 40]}
graph{(y3x8)(x+2)+14=0 [80, 80, 40, 40]}
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To find the vertical, horizontal, and oblique asymptotes for (\frac{3x^2 + 14x + 4}{x + 2}), follow these steps:

Vertical Asymptotes: Set the denominator equal to zero and solve for (x). Any (x) values obtained will be the vertical asymptotes.
(x + 2 = 0)
(x = 2)
So, the vertical asymptote is (x = 2).

Horizontal Asymptote: If the degree of the numerator is equal to or less than the degree of the denominator, divide the leading coefficients to find the horizontal asymptote.
In this case, the degree of the numerator ((2)) is less than the degree of the denominator ((1)). So, there is a horizontal asymptote.
The horizontal asymptote is given by the ratio of the leading coefficients of the numerator and denominator.
Horizontal asymptote: (y = \frac{3}{1} = 3)

Oblique Asymptote: If the degree of the numerator is exactly one more than the degree of the denominator, perform polynomial long division to find the oblique asymptote.
In this case, the degree of the numerator ((2)) is one more than the degree of the denominator ((1)). So, there might be an oblique asymptote.
Perform polynomial long division or use synthetic division to divide (3x^2 + 14x + 4) by (x + 2) to get the oblique asymptote.
The oblique asymptote can be found to be (y = 3x + 8).
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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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