How do you find the vertical, horizontal or slant asymptotes for #y=(2x)/(x-3)#?

Answer 1

#color(blue)(x=3)# is a Vertical Asymptote.

#color(blue)(y=2)# is the Horizontal Asymptote.

#No Slant Asymptote#

Vertical Asymptote is determined by setting the denominator to zero :

#x-3=0rArrx=3# Therefore ,
#color(blue)(x=3)# is a Vertical Asymptote.

The degree of the numerator is the same as that of the denominator , so there is Horizontal Asymptote but no slant Asymptote .

If the numerator and denominator have the same degree

#(a x^ n +b x +c )/(a 'x ^n + b')#
Then the Horizontal Asymptote is , #a/(a')" #,the fraction formed by their coefficients of the highest degree.
In the given quotient ,the numerator and denominator have the same degree #1# ,

therefore,

#color(blue)(y=2)# is the Horizontal Asymptote.
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Answer 2

To find the vertical asymptote, set the denominator equal to zero and solve for x. For the function y = (2x)/(x - 3), the vertical asymptote occurs at x = 3.

To find the horizontal asymptote, 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 y = 0. If the degrees are equal, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote. In this case, since the degree of the numerator (1) is less than the degree of the denominator (1), the horizontal asymptote is y = 0.

To find the slant asymptote, divide the numerator by the denominator using polynomial long division or synthetic division. If the result is a linear function, that's the equation of the slant asymptote. If not, there is no slant asymptote. For y = (2x)/(x - 3), the result of the division is 2, which is a constant. Therefore, there is no slant asymptote.

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Answer 3

To find the vertical asymptote, set the denominator equal to zero and solve for x. In this case, x - 3 = 0, so x = 3 is the vertical asymptote. There is no horizontal asymptote for this function. To find a slant asymptote, perform polynomial long division between the numerator and the denominator. The slant asymptote equation can be determined from the quotient obtained during the division.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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