How do you find the vertical, horizontal or slant asymptotes for #f(x)=(x3 )/ (2x1 )#?
vertical asymptote
horizontal asymptote
Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation let the denominator equal zero.
divide all terms on numerator/denominator by x
Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes.
Here is the graph of f(x). graph{(x3)/(2x1) [10, 10, 5, 5]}
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To find the vertical, horizontal, or slant asymptotes for ( f(x) = \frac{x  3}{2x  1} ):

Vertical Asymptotes: Vertical asymptotes occur where the denominator of the rational function equals zero and the numerator does not. So, set the denominator ( 2x  1 ) equal to zero and solve for ( x ).
( 2x  1 = 0 )
( 2x = 1 )
( x = \frac{1}{2} )Therefore, the vertical asymptote is at ( x = \frac{1}{2} ).

Horizontal Asymptotes: To find horizontal asymptotes, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ( y = 0 ). If the degrees are equal, divide the leading coefficients to find the horizontal asymptote.
In this case, the degrees are equal (both 1). So, divide the leading coefficients.
( \frac{1}{2} )
Therefore, the horizontal asymptote is at ( y = \frac{1}{2} ).

Slant Asymptotes: Slant asymptotes occur when the degree of the numerator is one greater than the degree of the denominator. To find the slant asymptote, perform polynomial long division.
( (x  3) ) divided by ( (2x  1) ):
( x  3 ) = ( \frac{1}{2} )(( 2x  1 )) + ( \frac{1}{2} )
So, the quotient is ( \frac{1}{2}x + \frac{1}{2} ).
Therefore, the slant asymptote is ( y = \frac{1}{2}x + \frac{1}{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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