How do you find the vertical, horizontal and slant asymptotes of: #f(x)=sinx/(x(x^281))#?
It has a hole at
It has no slant asymptotes.
Given:
Note that:
Footnote
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To find the vertical, horizontal, and slant asymptotes of the function ( f(x) = \frac{\sin x}{x(x^2  81)} ):

Vertical Asymptotes: These occur where the denominator equals zero, but the numerator does not. Set the denominator equal to zero and solve for ( x ).

Horizontal Asymptotes: To find horizontal asymptotes, we examine the behavior of the function as ( x ) approaches positive or negative infinity.

Slant Asymptotes: Slant asymptotes occur when the degree of the numerator is one greater than the degree of the denominator. We perform polynomial long division to check for a slant asymptote. If the result is a nonzero constant, it's a horizontal line which serves as the slant asymptote.
Performing these steps:

Vertical Asymptotes: The function has vertical asymptotes at ( x = 0 ) and ( x = ±9 ) because these are the points where the denominator becomes zero.

Horizontal Asymptotes: As ( x ) approaches positive or negative infinity, the function behaves like ( \frac{\sin x}{x^3} ), which approaches zero. Therefore, there is a horizontal asymptote at ( y = 0 ).

Slant Asymptotes: Perform polynomial long division of ( \sin x ) by ( x(x^2  81) ) to see if there's a slant asymptote. The result will give the slant asymptote if it's a nonzero constant. However, in this case, the division yields zero, indicating no slant asymptote.
So, summarizing:
 Vertical asymptotes: ( x = 0, \pm 9 )
 Horizontal asymptote: ( y = 0 )
 No slant asymptote.
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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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