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

2 Answers
Bill K.
Oct 24, 2015

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.

Truong-Son N.
Jul 27, 2016

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)#

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