The value of _ x → 0 (1 + 2x + 4x^ 2 ) + (1 - 2x + 4x^ 2 ) sec x -...

\lim_{x \to 0}\dfrac{\ln \left(1 + 2x + 4x^{2}\right) + \ln \left(1 - 2x + 4x^{2}\right)}{sec x - cos x} = \lim_{x \to 0}\dfrac{\ln \left[{\left(1 + 4x^{2}\right)}^{2} - 4x^{2}\right]}{\left(1 - cos^{2}x\right) / cos x} = \lim_{x \to 0}\dfrac{\ln \left(1 + 4x^{2} + 16x^{4}\right)}{sin xtan x} = \lim_{x \to 0}\dfrac{\ln \left(1 + 4x^{2}\left(1 + 4x^{2}\right)\right)}{4x^{2}\left(1 + 4x^{2}\right)}4x^{2}\left(1 + 4x^{2}\right) ⋅ \dfrac{1}{\dfrac{sin x}{x} ⋅ \dfrac{tan x}{x} ⋅ x^{2}} = 4 \left(as\lim_{x \to 0}\dfrac{\ln \left(1 + x\right) }{x} =1\right)