If the function f(x) = \ cases _ e (1 - x + x ^ 2 ) + _ e (1 + x +...

Given,

f\left(x\right) = \left\{\begin{cases}\dfrac{\log_{e}\left(1 - x + {x}^{2}\right) + \log_{e}\left(1 + x + {x}^{2}\right)}{\sec x - \cos x}, & x \in \left(\dfrac{- \pi}{2}, \dfrac{\pi}{2}\right) - \left\{0\right\} \\ k & , x = 0\end{cases}\right)</p><p>⇒ \lim_{x \to 0}\dfrac{\left(\ln \left(1 + {x}^{2} + {x}^{4}\right)\right)\cos x}{1 - \cos^{2}x} = k$</p><p>Taking L.H.S$= \lim_{x \to 0}\dfrac{\left(\ln \left(1 + {x}^{2} + {x}^{4}\right)\right)\cos x}{1 - \cos^{2}x}$</p><p>$= \lim_{x \to 0}\dfrac{\left(\dfrac{\ln \left(1 + {x}^{2} + {x}^{4}\right)}{{x}^{2} + {x}^{4}}\right){x}^{2}\left(1 + {x}^{2}\right)\cos x}{\left(\dfrac{\sin^{2}x}{{x}^{2}}\right){x}^{2}}$</p><p>$= \lim_{x \to 0}\dfrac{\left(\dfrac{\ln \left(1 + {x}^{2} + {x}^{4}\right)}{{x}^{2} + {x}^{4}}\right)\left(1 + {x}^{2}\right)\cos x}{\left(\dfrac{\sin^{2}x}{{x}^{2}}\right)}$</p><p>$= \dfrac{\left(1\right)\left(1 + 0\right)1}{\left(1\right)} = 1$</p><p>Now equating with R.H.S we get,$k = 1