How do you identify all asymptotes or holes for #g(x)=(x^26x+8)/(x+2)#?
Given:
Divide the numerator by the denominator by grouping:
graph{(y(x^26x+8)/(x+2))(yx+8) = 0 [79.6, 80.4, 45.64, 34.36]}
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To identify asymptotes or holes for ( g(x) = \frac{{x^2  6x + 8}}{{x + 2}} ), we need to analyze its behavior as ( x ) approaches certain values.

Vertical Asymptote:
 A vertical asymptote occurs where the denominator of the rational function equals zero but the numerator does not.
 Set the denominator ( x + 2 ) equal to zero and solve for ( x ).
 ( x + 2 = 0 ) gives ( x = 2 ).
 Therefore, there is a vertical asymptote at ( x = 2 ).

Horizontal Asymptote:
 To find horizontal asymptotes, examine the behavior of the function as ( x ) approaches positive or negative infinity.
 If the degrees of the numerator and denominator are the same, the horizontal asymptote occurs at the ratio of the leading coefficients.
 Since both the numerator and denominator are of degree 1, the horizontal asymptote is given by the ratio of the leading coefficients, which is ( \frac{1}{1} = 1 ).
 Therefore, there is a horizontal asymptote at ( y = 1 ).

Hole:
 If factors in the numerator and the denominator cancel out, there's a hole in the graph.
 Factorize the numerator ( x^2  6x + 8 ).
 ( x^2  6x + 8 = (x  4)(x  2) ).
 There's no factor of ( x + 2 ) in the numerator, so there's no cancellation.
 Therefore, there are no holes in the graph of ( g(x) ).
In summary, the rational function ( g(x) = \frac{{x^2  6x + 8}}{{x + 2}} ) has a vertical asymptote at ( x = 2 ) and a horizontal asymptote at ( y = 1 ), but it does not have any holes.
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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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