Question

(X-27)/[(X^1/3)-3] find the limit as X approaches 27

Answer 1

use\[x^3-a^3=(x-a)(x^2+xa+a^2)\]

Answer 2

replace \[x\] with \[x^{1/3}\]

Answer 3

and what about a?

Answer 4

a=3

Answer 5

\[x-27=(x^{1/3}-3)(x^{2/3}+x^{1/3}3+9)\]

Answer 6

the first bracket simplifies to zero no?

Answer 7

\[\frac{x-27}{x^{1/3}-3}=\frac{(x^{1/3}-3)(x^{2/3}+x^{1/3}3+9)}{x^{1/3}-3}\]

\[=\frac{\cancel{(x^{1/3}-3)}(x^{2/3}+x^{1/3}3+9)}{\cancel{x^{1/3}-3}}=x^{2/3}+x^{1/3}3+9\]

Now take limit...