Question

What is the limit of `((5u^4)+8)/((u^2)-5)((7u^2)-1)` as x goes to infinity?

Answer 1

It is not clear what the intended question is.

Explanation:

I shall assume that the variables should not be `u` and `x`
(if they should be different, there is not enough information to determine the limit.)

`lim_(urarroo)((5u^4)+8)/((u^2)-5)((7u^2)-1) = lim_(urarroo)(((5u^4)+8)((7u^2)-1))/((u^2)-5)`

`= lim_(xrarroo)(35u^6-5u^4+56u^2-8)/(u^2-5) = oo`

(The degree of the numerator is greater than the degree of the denominator, so the limit is either `oo` or `-oo`.)

If the intended limit is `lim_(xrarroo) (5x^4+8)/((x^2-5)(7x^2-1))`

Then we have:

`lim_(xrarroo) (5x^4+8)/((x^2-5)(7x^2-1)) = lim_(xrarroo) (5x^4+8)/(7x^4-36x^2+5)`

`= (x^4(5+8/x^4))/(x^4(7-36/x^2+5/x^4)) = 5/7`