How do you find vertical, horizontal and oblique asymptotes for #1/(x^22x8)#?
The vertical asymptotes are
The horizontal asymptote is
No oblique asymptote
Let's factorise the denominator
graph{(y1/(x^22x8))(y10000(x4))(y1000(x+2))(y)=0 [3.85, 7.25, 2.893, 2.66]}
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To find the vertical, horizontal, and oblique asymptotes for the function (f(x) = \frac{1}{x^2  2x  8}), follow these steps:

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). Any (x) value obtained is a potential vertical asymptote.
For (f(x) = \frac{1}{x^2  2x  8}), the denominator (x^2  2x  8) factors into ((x  4)(x + 2)). Setting each factor equal to zero, we find that (x = 4) and (x = 2) are vertical asymptotes.

Horizontal Asymptotes: Horizontal asymptotes occur when (x) approaches positive or negative infinity. To find horizontal asymptotes, analyze the behavior of the function as (x) approaches positive and negative infinity.
As (x) approaches positive or negative infinity, the terms with the highest degree in the numerator and denominator dominate the function. Since both the numerator and denominator are quadratic, divide both by (x^2) and take the limit as (x) approaches infinity.
[ \lim_{x \to \pm \infty} \frac{1}{x^2  2x  8} = 0 ]
This means the horizontal asymptote is (y = 0).

Oblique Asymptotes: Oblique asymptotes occur when the degree of the numerator is exactly 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.
[ \frac{1}{x^2  2x  8} = \frac{0}{x  4} + \frac{1}{x + 2} ]
As (x) approaches positive or negative infinity, the term (\frac{0}{x  4}) tends to zero, leaving the oblique asymptote as (y = \frac{1}{x + 2}).
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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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