Question

lim (cubic root (7+x^2-8x^3)/(x^3-x+pi))
x--> +inf

I don't know how to take the limit when there is a cubic root around . Between the ansswer is + inf . I need direction

Answer 1

as x gets big the numerator is dominated by the
-8 x^3 term

as x gets big the denominator is dominated by the x^3 term

so the numerator over the denominator
----> -8 x^3/x^3 = -8

Answer 2

what about the cubic root what do we have to do with it ?

Answer 3

Oh, did not notice that.
(-8)^(1/3)

that is a complex number
e^iT = cos T + i sin T

-8 = 8 cos pi + 8 sin pi
= 8 e^pi i

(8 e^i pi)^(1/3)

= 2 e^ipi/3 or 2 e^i(3pi/3) or 2e^i(5 pi/3)

= 2(cos 60+ i sin 60)
or
= 2 (cos 180 + i sin 180)
or
= 2 (cos 300 + i sin 300)

= 1 + 1.73 i
or
= -2

or
= -1 - 1.73 i

so
-2 if only doing real numbers