Question

What is the limit of `[3 + 4/x - 5/x^2 + [x-1]/[x^3+1]` as x goes to infinity?

Answer 1

3

Explanation:

Every term in the expression `3+4/x-5/(x^2)+(x-1)/(x^3+1)` goes to zero as `x rightarrow infty` except the 3.

The reason is that the other terms are rational functions where the denominator has a higher degree than the numerator.

Answer 2

If we actually took the limit, we can separate each term. Since each of these functions have existent limits...

`color(blue)(lim_(x->oo) 3 + 4/x - 5/x^2 + (x-1)/(x^3+1))`

`= lim_(x->oo) 3 + lim_(x->oo) 4/x - lim_(x->oo) 5/x^2 + lim_(x->oo) (x-1)/(x^3+1)`

As `x->oo` for `(x-1)/(x^3+1)`, the `-1` and `+1` become insignificant, so this is equivalent to `lim_(x->oo) 1/(x^2)`:

`=> 3 + cancel(4/(oo))^(0) - cancel(5/(oo))^(0) + cancel(lim_(x->oo) 1/(x^2))^(1/(oo) -> 0)`

`= color(blue)(3)`