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

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Answer 1 · 2015-10-24T14:48:15

3

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 · 2016-07-27T22:33:21

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 $lim_(x->oo) 1/(x^2)$ :

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

$= color(blue)(3)$

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