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How do you find the limit of # (3 x^4 + 4) / ((x^2 - 7)(4 x^2 - 1)) # as x approaches infinity?

User MJPinfield
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\displaystyle\lim_(x\to\infty)(3x^4+4)/((x^2-7)(4x^2-1))

Expanding the denominator gives



(x^2-7)(4x^2-1)=4x^4-29x^2+7

Then in the whole rational expression, we can divide through by
x^4. Since
x\to\infty, we assume that
x>0, so this is legal.


(3x^4+4)/(4x^4-29x^2+7)=\frac{3+\frac4{x^4}}{4-(29)/(x^2)+\frac7{x^4}}

As
x gets arbitrarily large, the rational terms in the numerator and denominator become negligible. So for large
x, we have


(3x^4+4)/((x^2-7)(4x^2-1))\approx\frac34

And so the limit is
\frac34.
User Nikolay Prokopyev
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