How do you identify all asymptotes or holes for #y=(x^31)/(x^2+2x)#?
There is no hole. Vertical asymptotes are
graph{(x^31)/(x^2+2x) [41.33, 38.67, 22.32, 17.68]}
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To identify the asymptotes or holes for the function ( y = \frac{x^3  1}{x^2 + 2x} ), follow these steps:

Identify Vertical Asymptotes: Vertical asymptotes occur where the denominator equals zero and the numerator doesn't simultaneously. Set the denominator ( x^2 + 2x ) equal to zero and solve for ( x ). The values of ( x ) obtained are the vertical asymptotes unless they are also roots of the numerator, in which case there could be holes. [ x^2 + 2x = 0 ] [ x(x + 2) = 0 ] [ x = 0 \quad \text{or} \quad x = 2 ]

Check for Holes: Substitute the values of ( x ) obtained above into the function. If any of them make the numerator zero, then there is a hole at that point. If they don't, then there is no hole.
 Substitute ( x = 0 ) into ( \frac{x^3  1}{x^2 + 2x} ): [ \frac{(0)^3  1}{(0)^2 + 2(0)} = 1 ] Since the numerator is not zero, there is no hole at ( x = 0 ).
 Substitute ( x = 2 ) into ( \frac{x^3  1}{x^2 + 2x} ): [ \frac{(2)^3  1}{(2)^2 + 2(2)} = \frac{9}{0} ] Since the numerator is not zero, there is a vertical asymptote at ( x = 2 ).

Identify Horizontal Asymptotes: Horizontal asymptotes can occur when the degree of the numerator is less than, equal to, or greater than the degree of the denominator. In this case, since the degree of the numerator is one greater than the degree of the denominator, there is no horizontal asymptote.
Therefore, the identified asymptotes are:
 Vertical asymptote: ( x = 2 )
 There are no 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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