$ \displaystyle\lim_{x \to 0} \frac{tan x}{x} = \displaystyle\lim_{x \to 0} \frac{sin x}{cos x} \frac{1}{x} =\displaystyle\lim_{x \to 0} \frac{sin x}{x} \frac{1}{cos x} = 1\cdot 1 = 1 $
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