Question

${lim}_{x\to 0}\frac{\sqrt[5]{52+x}-\sqrt[5]{52-x}}{\sin hx}$=

• A. 1
• B. $\sqrt[5]{52}$
• C. $\frac{\sqrt{5}}{2}$
• D. $\frac{2^{\frac{1}{5}}}{5}$

Answer 1

Answer: (D)

$\frac{2^{\frac{1}{5}}}{5}$

To find ${lim}_{x\to 0}\frac{\sqrt[5]{52+x}-\sqrt[5]{52-x}}{\sin hx}$,

Use L-Hospital rule, since it's in $\frac{0}{0}$ form. Find the derivative of both the numerator and the denominator

${lim}_{x\to 0}\frac{\sqrt[5]{52+x}-\sqrt[5]{52-x}}{\sin hx}={lim}_{x\to 0}\frac{\frac{1}{5.\sqrt[5]{5{(2+x)}^4}}+\frac{1}{5.\sqrt[5]{5{(2-x)}^4}}}{\cosh x}$

Now, substitute $x=0$ to find the value of the limit

$=\frac{2^{\frac{1}{5}}}{5}$

Option D is correct.