Question

The value of $\underset{x\to 0}{lim}\frac{{(1+x)}^{1/4}-{(1-x)}^{1/4}}{x}$ is

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

Answer 1

The correct option is A $\frac{1}{2}$

$\underset{x\to 0}{lim}\frac{{(1+x)}^{\frac{1}{4}}-{(1-x)}^{\frac{1}{4}}}{x}$

$=\underset{x\to 0}{lim}\frac{{(1+x)}^{\frac{1}{2}}-{(1-x)}^{\frac{1}{2}}}{x({(1+x)}^{\frac{1}{4}}+{(1-x)}^{\frac{1}{4}})}$ (Rationalizing )

$=\underset{x\to 0}{lim}\frac{(1+x)-(1-x)}{x({(1+x)}^{\frac{1}{4}}+{(1-x)}^{\frac{1}{4}})({(1+x)}^{\frac{1}{2}}+{(1-x)}^{\frac{1}{2}})}$ (Again rationalizing )

$=\underset{x\to 0}{lim}\frac{2}{({(1+x)}^{\frac{1}{4}}+{(1-x)}^{\frac{1}{4}})({(1+x)}^{\frac{1}{2}}+{(1-x)}^{\frac{1}{2}})}=\frac{1}{2}$

Hence, option 'A' is correct.