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Ta có $cos\sqrt{x}$ $ \sim$ $ 1 -\frac{x}{2}$ $( x \rightarrow 0).$ Do đó $\mathop {\lim }\limits_{x \to 0} \sqrt[x]{cos\sqrt{x}}=$$\mathop {\lim }\limits_{x \to 0}(1-\frac{x}{2})^{\frac{1}{x}}$$ =\mathop {\lim }\limits_{x \to 0}(1-\frac{x}{2})^{\frac{-2}{x}.\frac{1}{-2}} =\frac{1}{\sqrt{e}}$.
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