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lim-x-0-x-2-cos-1-x-




Question Number 119482 by liberty last updated on 24/Oct/20
 lim_(x→0)  x^2  cos ((1/x)) ?
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}^{\mathrm{2}} \:\mathrm{cos}\:\left(\frac{\mathrm{1}}{{x}}\right)\:? \\ $$
Answered by Olaf last updated on 25/Oct/20
∀ x∈R^∗ , −1 ≤ cos((1/x)) ≤ +1  ∀ x∈R^∗ , −x^2  ≤ x^2 cos((1/x)) ≤ +x^2   And lim_(x→0) (±x^2 ) = 0  ⇒ lim_(x→0)  x^2 cos((1/x)) = 0
$$\forall\:{x}\in\mathbb{R}^{\ast} ,\:−\mathrm{1}\:\leqslant\:\mathrm{cos}\left(\frac{\mathrm{1}}{{x}}\right)\:\leqslant\:+\mathrm{1} \\ $$$$\forall\:{x}\in\mathbb{R}^{\ast} ,\:−{x}^{\mathrm{2}} \:\leqslant\:{x}^{\mathrm{2}} \mathrm{cos}\left(\frac{\mathrm{1}}{{x}}\right)\:\leqslant\:+{x}^{\mathrm{2}} \\ $$$$\mathrm{And}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\pm{x}^{\mathrm{2}} \right)\:=\:\mathrm{0} \\ $$$$\Rightarrow\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}^{\mathrm{2}} \mathrm{cos}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{0} \\ $$

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