How-would-you-find-the-limit-of-lim-x-x-3-x-1-3-

Question Number 6405 by FilupSmith last updated on 26/Jun/16

How would you find the limit of :

\lim_{x \to \infty} x^{3} - {(x - 1)}^{3}

Commented by FilupSmith last updated on 26/Jun/16

Do you have to expand and simplify ?

Or is there a simpler method ?

Commented by Rasheed Soomro last updated on 27/Jun/16

\lim_{x \to \infty} x^{3} - {(x - 1)}^{3} = ?

x^{3} - {(x - 1)}^{3}

= x^{3} - x^{3} + 3 x^{2} - 3 x + 1

= 3 x^{2} - 3 x + 1

= 3 x^{2} - 6 x + 3 + 3 x - 2

= 3 (x^{2} - 2 x + 1) + 3 x - 2

= 3 {(x - 1)}^{2} + 3 x - 2

\lim_{x \to \infty} x^{3} - {(x - 1)}^{3} = \lim_{x \to \infty} (3 {(x - 1)}^{2} + 3 x - 2)

= 3 {(\infty - 1)}^{2} + 3 (\infty) - 2 = \infty

Commented by Rasheed Soomro last updated on 27/Jun/16

\lim_{x \to \infty} x^{3} - {(x - 1)}^{3} = ?

x^{3} - {(x - 1)}^{3}

= (x - (x - 1)) (x^{2} + x (x - 1) + {(x - 1)}^{2})

= x^{2} + x (x - 1) + {(x - 1)}^{2}

\lim_{x \to \infty} x^{3} - {(x - 1)}^{3} = \lim_{x \to \infty} (x^{2} + x (x - 1) + {(x - 1)}^{2})

\infty^{2} + \infty (\infty - 1) + {(\infty - 1)}^{2}

= \infty + \infty + \infty = \infty

Commented by Rasheed Soomro last updated on 27/Jun/16

{D o n}^{'} t k n o w s i m p l e r m e t h o d .