What is the limit of #root3[x^3+2] - root3[x^3-1]# as x goes to infinity?

1 Answer
Sep 26, 2015

#lim_(xrarroo)(root3[x^3+2] - root3[x^3-1])=0#

Explanation:

Use the difference of cubes to rewrite the expression.

#(a-b)(a^2+ab+b^2) = a^3-b^3#

So,

#(root3a-root3b)(root3a^2+root3(ab)+root3b^2) = a-b#

And, so

#(root3a-root3b)/1 * (root3a^2+root3(ab)+root3b^2)/(root3a^2+root3(ab)+root3b^2) = (a-b)/(root3a^2+root3(ab)+root3b^2)#

In this case, we have:

#root3[x^3+2] - root3[x^3-1] = ((x^3+2)-(x^3-1))/(root3(x^3+2)^2+root3((x^3+2)(x^3-1))+root3(x^3-1)^2)#

# = 3/(root3(x^3+2)^2+root3((x^3+2)(x^3-1))+root3(x^3-1)^2)#

As #x# increases without bound, the denominator also increases without bound, so the limit of the ratio is #0#.