How do you find the asymptotes for #f(x)=(3x^5 + 1) / (2x^6 + 3x 1)#?
#x_1 ~~ 0.3324335502431692846#
#x_2 ~~ 1.1414892917449508403#
graph{(3x^5+1)/(2x^6+3x1) [10, 10, 5, 5]}
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To find the asymptotes for the function ( f(x) = \frac{{3x^5 + 1}}{{2x^6 + 3x  1}} ):

Horizontal Asymptote:
 Compare the degrees of the numerator and denominator polynomials. In this case, the degree of the numerator is 5, and the degree of the denominator is 6.
 If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at y = 0.
 If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients. Here, the leading coefficient of the numerator is 3 and the leading coefficient of the denominator is 2. So, the horizontal asymptote is at ( y = \frac{3}{2} ).

Vertical Asymptotes:
 Set the denominator equal to zero and solve for x. Any values of x that make the denominator zero will give vertical asymptotes.
 Solve ( 2x^6 + 3x  1 = 0 ) for x. This equation may require numerical methods to solve since it is a sixthdegree polynomial equation. Once you find the values of x, those are the vertical asymptotes.
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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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