If the function f(x)= c _ e (1-x+x^ 2 )+ _ e (1+x+x^ 2 ) x- x , x… - Vidyadip

MCQ

If the function $f(x)=\left\{\begin{array}{c}\frac{\log _{e}\left(1-x+x^{2}\right)+\log _{e}\left(1+x+x^{2}\right)}{\sec x-\cos x}, x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)-\{0\} \\ k \end{array}\right.$ is continuous at $x =0$, then $k$ is equal to.

✓
$1$
B
$-1$
C
$e$
D
$0$

✓

Answer

Correct option: A.

$1$

a
$\lim _{x \rightarrow 0} \frac{\left(\ln \left(1+x^{2}+x^{4}\right)\right) \cos x}{1-\cos ^{2} x}$

$\lim _{x \rightarrow 0} \frac{\left(\frac{\ln \left(1+x^{2}+x^{4}\right)}{x^{2}+x^{4}}\right) x^{2}\left(1+x^{2}\right) \cos x}{\left(\frac{\sin ^{2} x}{x^{2}}\right) x^{2}}=1$

$\therefore k =1$

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