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Question Number 55639 by gunawan last updated on 01/Mar/19

$$\mathrm{Known}a\in \mathbb{R}\mathrm{and}$$ $$\mathrm{function}f:\mathbb{R}\to \mathbb{R}\mathrm{satiesfied}$$ $$\mid xf\left(x\right)+a\mid <{\sin }^2(x-a).$$ $$\mathrm{For}\mathrm{all}x\in \mathbb{R}$$ $$\mathrm{value}\mathrm{of}\underset{x\to a}{\lim }f\left(x\right)..$$

Answered by kaivan.ahmadi last updated on 01/Mar/19

$$-{sin}^2(x-a)<xf\left(x\right)+a<{sin}^2(x-a)$$ $$bysandwichtheorem$$ $$li\underset{x\to a}{mxf}\left(x\right)+a=0\Rightarrow li\underset{x\to a}{m}xf\left(x\right)=-a\Rightarrow$$ $$li\underset{x\to a}{mx}\times li\underset{x\to a}{mf}\left(x\right)=-a\Rightarrow$$ $$li\underset{x\to a}{mf}\left(x\right)=-1$$

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