How do you find vertical, horizontal and oblique asymptotes for #f(x) = (4x)/(x^2-1)#?

Answer 1

vertical asymptotes x = ± 1
horizontal asymptote y = 0

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation/s equate the denominator to zero.

solve : # x^2 - 1 = 0 → (x-1)(x+1) = 0 #
# rArr x = ± 1" are the asymptotes " #
Horizontal asymptotes occur as # lim_(x→±∞) f(x) → 0 #

If the degree of the numerator is less than the degree of the denominator , as is the case here then equation is y = 0 .

Oblique asymptotes occur when the degree of the numerator is greater than the degree of the denominator.This is not the case here and so there is no oblique asymptote.

Here is the graph of the function. graph{4x/(x^2-1) [-10, 10, -5, 5]}

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Answer 2

To find the vertical asymptotes, determine the values of ( x ) for which the denominator ( x^2 - 1 ) equals zero. These values are ( x = 1 ) and ( x = -1 ). So, the vertical asymptotes are ( x = 1 ) and ( x = -1 ).

To find the horizontal asymptote, compare the degrees of the numerator and denominator polynomials. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ( y = 0 ).

To find the oblique asymptote, divide the numerator polynomial by the denominator polynomial using polynomial 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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