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I know, how to calculate $$ \lim_{x\to0}\frac{\cos x-1}{x^2} $$ without differential calculus. Calculating $$ \lim_{x\to0}\frac{\sin x-x}{x^3} $$ using de l'Hospital's rule or Taylor expansion is also easy.

Is there a method to calculate the previous limit without de l'Hospital's rule or stronger tools?

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    $\begingroup$ The solution is provided by the second answer (user17762's one) given here: math.stackexchange.com/questions/134051/… $\endgroup$ – Marco Vergura Dec 26 '14 at 11:14
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    $\begingroup$ Thank you both! It is a trick I need. $\endgroup$ – Przemysław Scherwentke Dec 26 '14 at 11:18
  • $\begingroup$ @PrzemysławScherwentke I see you have created new tag called (elementary-proof). It is recommended that a user who creates a new tag also creates tag-info - at least some basic info on the intended use of the new tag in the tag-excerpt. See, for example, this answer on meta. If some more details are needed, feel free to drop me a line in chat. $\endgroup$ – Martin Sleziak Dec 27 '14 at 10:06