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Question Number 126378 by benjo_mathlover last updated on 20/Dec/20

$$\underset{x\to 0}{\lim }\frac{\cos x-\cos 3x+x^3\cos \left(\frac{\pi }{x}\right)}{x^2}?$$

Answered by liberty last updated on 20/Dec/20

$$\underset{x\to 0}{\lim }\frac{\cos x-\cos 3x}{x^2}+\underset{x\to 0}{\lim }\frac{x^3\cos \left(\frac{\pi }{x}\right)}{x^2}=$$$$\underset{x\to 0}{\lim }\frac{(1-\frac{1}{2}{x}^2)-(1-\frac{9}{2}{x}^2)}{x^2}+\underset{x\to 0}{\lim }x\cos \left(\frac{\pi }{x}\right)=$$$$\underset{x\to 0}{\lim }\frac{4x^2}{x^2}+\underset{x\to 0}{\lim }x\cos \left(\frac{\pi }{x}\right)=4+0=4$$

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