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

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

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

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

To evaluate the given limit, let's first simplify the expression inside the limit:

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

As x approaches positive infinity, let's focus on the highest power terms in the numerator and denominator. In this case, the highest power is x^3 in both the numerator and denominator.

Now, divide both the numerator and denominator by x^3:

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

Simplifying further:

(cubic root (7/x^3 + 1/x - 8)/(1 - 1/x^2 + pi/x^3))

As x approaches infinity, terms with lower power will approach zero. Therefore, the terms 1/x and 1/x^3 will approach zero, and the expression can be simplified to:

(cubic root (1)/(1 - 0 + 0))

Now, we are left with:

(cubic root (1)/1)

Since the cubic root of 1 is equal to 1, the expression becomes:

1/1 = 1

Therefore, the limit as x approaches positive infinity of the given expression is 1.