Prove that cube root of (x^3 + 6) - cube root of (x^3 + 3) is approximately equal to 1/x^2 when x is sufficiently large.

Question

Prove that \(\sqrt[3]{x^3 + 6}\) - \(\sqrt[3]{x^3 + 3}\) is approximately equal to 1/x² when x is sufficiently large.

Answer 1

When x is large then 1/x will be small (i.e.) |1/x| < 1

So