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Question Number 131386 by bemath last updated on 04/Feb/21

$$\underset{x\to -\mathrm{\infty }}{\lim }\frac{x^5\cos \left(\frac{1}{\pi x^2}\right)+x^6\sin \left(\frac{1}{\pi x}\right)+7}{\mid x{\mid }^5+6\mid x\mid +7}=?$$

Answered by liberty last updated on 04/Feb/21

$$\underset{x\to -\mathrm{\infty }}{\lim }\frac{{\mathrm{x}}^5\cos \left(\frac{1}{\pi {\mathrm{x}}^2}\right)+{\mathrm{x}}^6\sin \left(\frac{1}{\pi \mathrm{x}}\right)+7}{\mid \mathrm{x}{\mid }^5+6\mid \mathrm{x}\mid +7}=?$$$$\mathrm{remark}\mid \mathrm{x}\mid =-\mathrm{x}\mathrm{as}\mathrm{x}\to -\mathrm{\infty }$$$$\underset{x\to -\mathrm{\infty }}{\lim }\frac{{\mathrm{x}}^5\cos \left(\frac{1}{\pi {\mathrm{x}}^2}\right)+{\mathrm{x}}^6\sin \left(\frac{1}{\pi \mathrm{x}}\right)+7}{-{\mathrm{x}}^5-6\mathrm{x}+7}$$$$=\underset{x\to -\mathrm{\infty }}{\lim }\frac{\cos \left(\frac{1}{\pi {\mathrm{x}}^2}\right)+\mathrm{x}\sin \left(\frac{1}{\pi \mathrm{x}}\right)+\frac{7}{{\mathrm{x}}^5}}{-1-\frac{6}{{\mathrm{x}}^4}+\frac{7}{{\mathrm{x}}^5}}$$$$[\mathrm{note}:\underset{x\to \mathrm{\infty }}{\lim }\mathrm{x}\sin \left(\frac{1}{\mathrm{x}}\right)=1]$$$$=\underset{x\to -\mathrm{\infty }}{\lim }\frac{\cos \left(\frac{1}{\pi {\mathrm{x}}^2}\right)+\mathrm{x}\sin \left(\frac{1}{\pi \mathrm{x}}\right)+\frac{7}{{\mathrm{x}}^5}}{-1-\frac{6}{{\mathrm{x}}^4}+\frac{7}{{\mathrm{x}}^5}}=-1-\frac{1}{\pi }$$

Commented by liberty last updated on 04/Feb/21

Commented by bramlexs22 last updated on 04/Feb/21

$$$$$$\mathrm{ruhig}\mathrm{und}\mathrm{khl}$$

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