Question

$\underset{x\to 0}{lim}\frac{\sqrt{1+x^3}-\sqrt{1-x^3}}{x^3}=$

• A. $0$
• B. $\frac{1}{2}$
• C. $1$
• D. $-\frac{1}{2}$

Answer 1

The correct option is C $1$

$\underset{x⟶0}{lim}\frac{\sqrt{1+x^3}-\sqrt{1-x^3}}{x^3}=\underset{x⟶0}{lim}\frac{\sqrt{1+x^3}-\sqrt{1-x^3}}{x^3}\times \frac{\sqrt{1+x^3}+\sqrt{1-x^3}}{\sqrt{1+x^3}+\sqrt{1-x^3}}=\underset{x⟶0}{lim}\frac{1+x^3-1+x^3}{x^3(\sqrt{1+x^3}+\sqrt{1-x^3})}=\underset{x⟶0}{lim}\frac{2}{\sqrt{1+x^3}+\sqrt{1-x^3}}=\frac{2}{1+1}=1$