Horizontal Asymptote of f(x) = 2x2/(x4-81)1/2 - How to Factor Bottom

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To find the horizontal asymptote of the function f(x) = 2x^2/(√(x^4 - 81)), the denominator can be factored as √(x^4(1 - 81/x^4)). As x approaches infinity, the function simplifies to f(x) ≈ 2x^2/√(x^4), which further reduces to 2/1, leading to the conclusion that the horizontal asymptote is y = 2. The limit of f(x) as x approaches large values in either direction confirms this result. Therefore, the horizontal asymptote for the given function is y = 2.
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Homework Statement


Find the horizontal asymptote to the graph: f(x) = 2x2/(x4-81)1/2


Homework Equations





How do I factor the bottom? Because for, the HA, I compare the coefficients.
 
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x4-81=x4(1-81/x4)

and remember that √(ab)=√a * √b
 
How do I prove that the HA: is y=2?
 
Look at
f(x)~=~\frac{2x^2}{\sqrt{x^4(1 - 81/x^4)}}
and simplify the denominator.

What is the limit of f(x) as x gets large in either direction?
 

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