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InterviewSolution
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Evaluate : \(\lim\limits_{x \to \frac{\pi}{4}}\frac{tan^3x-tanx}{cos(x+\frac{\pi}{4})}\) |
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Answer» \(\lim\limits_{x \to \frac{\pi}{4}}\frac{tan^3x-tanx}{cos(x+\frac{\pi}{4})}\) \(=\lim\limits_{x \to \frac{\pi}{4}}\frac{tanx(tan^2x-1)}{cos(x+\frac{\pi}{4})}\) \(=\lim\limits_{x \to \frac{\pi}{4}}tanx.\lim\limits_{x \to \frac{\pi}{4}}[\frac{-(1-tan^2\,x)}{cos(x+\frac{\pi}{4})}]\) \(=1\times[-\lim\limits_{x \to \frac{\pi}{4}}\frac{(1+tan\,x)(1-tan\,x)}{cos(x+\frac{\pi}{4})}]\) \(=\lim\limits_{x \to \frac{\pi}{4}}(1+tanx)\lim\limits_{x \to \frac{\pi}{4}}[\frac{cos\,x-sin\,x}{cos\,x.cos(x+\frac{\pi}{4})}]\) \(=-2\sqrt 2\times\) \(\lim\limits_{x \to \frac{\pi}{4}}\frac{cos(x+\frac{\pi}{4})}{cos\,x.cos(x+\frac{\pi}{4})}\) ∵ [cos x − sin x = \(\sqrt{2}\,cos\,(x+\frac{\pi}{4})]\) |
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