......nice ... ... ... calculus..... If lim_(x→0) ((tan(x))/x) = 1 , prove that: lim(1/x)((1/x)−(1/(tan(x))))=(1/3)


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Question Number 141685 by mnjuly1970 last updated on 22/May/21

$. . . . . . n i c e . . . . . . . . . c a l c u l u s . . . . .$ $I f {l i m}_{x \to 0} \frac{t a n (x)}{x} = 1, p r o v e t h a t :$ $l i m \frac{1}{x} (\frac{1}{x} - \frac{1}{t a n (x)}) = \frac{1}{3}$
	Answered by iloveisrael last updated on 22/May/21

	$\lim_{x \to 0} \frac{\tan x - x}{x^{2} \tan x} =$ $\lim_{x \to 0} \frac{(x + \frac{1}{3} x^{3}) - x}{x^{2} (x + \frac{1}{3} x^{3})} =$ $\lim_{x \to 0} \frac{\frac{1}{3} x^{3}}{x^{3} (1 + \frac{1}{3} x^{2})} = \frac{1}{3}$
	Commented by greg_ed last updated on 23/May/21

	$true!$
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