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### Calculate the limit : $\lim_{x \to 0}\frac{x-\sin{x}}{x^3}$ WITHOUT using L'Hopital's rule [duplicate]

I was given a task to find $$\lim_{x\to0}\frac{x-\sin{x}}{x^3}$$ at my school today. I thought it was an easy problem and started differentiating denominator and numerator to calculate the limit but ...
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### $\lim_{x\to 0} \frac{\sin x - x}{x^2}$ without L'Hospital or Taylor

It is easy to see that $$\lim_{x\to 0} \frac{\sin x - x}{x^2} =0,$$but I can't figure out for the life of me how to argue without using L'Hospital or Taylor. Any ideas?
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### Trying to find $\lim_{x \to 0} \frac{x - \sin x}{(x \sin x)^{(3/2)}}$ using L'Hopital's

I'm trying to use L'Hopital's rule to calculate: $$\lim_{x \to 0^+} \dfrac{x - \sin x}{(x \sin x)^{(3/2)}}$$ Taking a couple of derivatives of the denominator gets quite nasty, so I'd like to find a ...
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### Find $\lim\limits_{x\to0}\frac{\sin(x)-\tan(x)}{\arcsin(x)-\arctan(x)}$ [duplicate]

Find $\displaystyle \lim_{x\to0}\frac{\sin(x)-\tan(x)}{\arcsin(x)-\arctan(x)}$ I tried using L'hopital's rule but it got very messy very fast UPDATE- So reading about the Taylor series this is what ...
Find the limit $$\lim_{x\rightarrow 0} \frac{\sin x-x}{\tan^3 x}$$ I found the limit which is $-\frac{1}{6}$ by using L'Hopital Rule. Is there another way to solve it without using the rule? Thanks ...