Question

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

Answer 1

$\displaystyle\lim_{x\to\infty}\frac{3x^4+4}{(x^2-7)(4x^2-1)}$

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.

$\dfrac{3x^4+4}{4x^4-29x^2+7}=\dfrac{3+\frac4{x^4}}{4-\dfrac{29}{x^2}+\dfrac7{x^4}}$

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

$\dfrac{3x^4+4}{(x^2-7)(4x^2-1)}\approx\dfrac34$

And so the limit is $\dfrac34$.