Find the limit as x->∞?

Question

Can somone please confirm my algebraic calculation of the following limit?

limx->∞ (x^(1/3)) / (1-x)

limx->∞ ((x^(1/3)) / x) / ((1-x) / x)

limx->∞ (1/x^(2/3)) / (1/x - 1)

= 0 / -1

= 0

I know the limit is zero, I am just making sure that my methods here are sound.

0 also results for limx->-∞ using this same process correct?

Thanks!

Answer 1

Looks sound to me.

Answer 2

Yay for calculus.

Anyways, the easiest way to test the answer is to plug in some numbers.

If this limit approached 8, it would be 2/(-7)
If this limit approached 27, it would be 3/(-26)
If this limit approached 64, it would be 4/ (-63)

So again, your answer is obviously right. I always use the shortcut: if power of x on top (1/3) is lower than on bottom (1), limit towards infinity is 0. The opposite is true but with a limit of infinity.

Answer 3

That looks good, I did it a different way and got 1/-infinity which is 0, by factoring out the x^(1/3).

Answer 4

Your methods are just fine. As you surmised, the same thing also happens as x→-∞.