How do you find the asymptotes for #f(x) = (2x^2 - 8) / (x^2 - 16)#?

Answer 1

vertical asymptotes at x = ± 4
horizontal asymptote at y = 2

Vertical asymptotes occur as the denominator of a rational function approaches zero. To find the equation let the denominator equal zero.

solve # x^2-16 = 0 #

This is a difference of squares and factors as (x-4)*x+4)=0

the equation of the vertical asymptotes are x = ± 4

[horizontal asymptotes occur as # lim_(x→±∞) f(x) → 0#]

If the degree of the denominator and numerator are equal then the equation can be found by taking the ratio of leading coefficients.

In this question they are both of degree 2

hence equation is # y = 2/1 = 2 #

Here is the graph of f(x) to illustrate these. graph{(2x^2-8)/(x^2-16) [-10, 10, -5, 5]}

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Answer 2

To find the asymptotes for the function ( f(x) = \frac{2x^2 - 8}{x^2 - 16} ):

  1. Vertical Asymptotes: Set the denominator equal to zero and solve for ( x ).

    ( x^2 - 16 = 0 )

    ( (x + 4)(x - 4) = 0 )

    ( x = -4 ) and ( x = 4 )

    So, there are vertical asymptotes at ( x = -4 ) and ( x = 4 ).

  2. Horizontal Asymptotes: Compare the degrees of the numerator and denominator.

    Degree of the numerator = 2

    Degree of the denominator = 2

    Since the degrees are equal, divide the leading coefficients to find the horizontal asymptote.

    Leading coefficient of numerator = 2

    Leading coefficient of denominator = 1

    So, the horizontal asymptote is ( y = \frac{2}{1} = 2 ).

  3. Oblique Asymptotes: Check if the degree of the numerator is one greater than the degree of the denominator.

    Since the degree of the numerator is not one greater than the degree of the denominator, there are no oblique asymptotes for this function.

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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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