How do you find the Vertical, Horizontal, and Oblique Asymptote given #y=1/(2-x)#?

Answer 1

vertical asymptote at x = 2
horizontal asymptote at y = 0

The denominator of y cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: #2-x=0rArrx=2" is the asymptote"#

Horizontal asymptotes occur as

#lim_(xto+-oo),ytoc" (a constant)"#

divide terms on numerator/denominator by x

#y=(1/x)/(2/x-x/x)=(1/x)/(2/x-1)#
as #xto+-oo,yto0/(0-1)#
#rArry=0" is the asymptote"#

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here ( numerator-degree 0 , denominator- degree 1 ) Hence there are no oblique asymptotes. graph{(1)/(2-x) [-10, 10, -5, 5]}

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Answer 2

To find the vertical asymptote of a function, set the denominator equal to zero and solve for x. In this case, set 2 - x equal to zero and solve for x.

To find the horizontal asymptote, analyze the behavior of the function as x approaches positive or negative infinity. In this case, observe that as x approaches positive or negative infinity, the function approaches 0.

To find the oblique asymptote, perform long division between the numerator and the denominator of the function. In this case, divide 1 by (2 - x) using long division. The oblique asymptote is the quotient obtained from this 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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