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Evaluate: \(\lim\limits_{x \to 0}\frac{sec4x-sec2x}{sec3x-secx}\)

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\(\lim\limits_{x \to 0}\frac{sec4x-sec2x}{sec3x-secx}\) 

\(=\lim\limits_{x \to 0}\frac {\frac{1}{cos4x}-\frac{1}{cos2x}}{\frac{1}{cos3x}-\frac{1}{cosx}}\) 

\(=\lim\limits_{x \to 0}\frac{\frac{cos2x-cos4x}{cos4x.cos2x}}{\frac{cosx-cos3x}{cos3x.cosx}}\) 

\(=\lim\limits_{x \to 0}(\frac{cos2x-cos4x}{cosx-cos3x}\times\frac{cosx.cos3x}{cos2x.cos4x})\) 

\(=\lim\limits_{x \to 0}(\frac{2sin3x.sinx}{2sin\,2x-sinx}\times\frac{cosx.cos3x}{cos2x.cos4x})\) 

\(=\lim\limits_{x \to 0}(\frac{sin3x}{sin2x}).\lim\limits_{x \to 0}(\frac{cosx.cos3x}{cos2x.cos4x})\) 

\(=\frac{3}{2}\frac{\lim\limits_{x \to 0}\frac{sin3x}{3x}}{\lim\limits_{x \to 0}\frac{sin2x}{2x}}\) 

\(=\frac{3}{2}.\frac{1}{1}\) 

\(=\frac{3}{2}\)

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