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Q. Let $f(x) = \begin{cases} \frac{\ln \left(8+x^{3}\right)-\ln \left(8-x^{3}\right)}{x^{3}}, & \text{if $x \neq 0$} \\[2ex] k, & \text{if $x=0$ } \end{cases}$
continuous at $x = 0$, then the value of k is ______.

Continuity and Differentiability

Solution:

$k=\displaystyle\lim_{x\to 0 } \frac{\ln \left(\frac{8+x^{3}}{8-x^{3}}\right)}{x^{3}}$
$=\displaystyle\lim_{x\to 0 } \frac{2 x^{3}}{\left(8-x^{3}\right) x^{3}}$
$=\frac{1}{4}$