Question

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

Answer 1

`lim_(x->+oo) (3x^4+4)/((x^2-7)(4x^2-1))=3/4`

Explanation:

Expand the denominator:

`(3x^4+4)/((x^2-7)(4x^2-1))= (3x^4+4)/(4x^4-29x^2+7)`

Now the limit for `x->oo` of a rational function only depends on the monomials of highest degree above and below the line.

As they are of the same degree, the limit is finite and equals the ratio of the leading coefficients:

`lim_(x->+oo) (3x^4+4)/((x^2-7)(4x^2-1))=3/4`