WWE2K14 ISO STASH GAMING.iso



 
 
 
 
 
 
 

WWE2K14 ISO STASH GAMING.iso

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How to prove this complicated limit?

How to prove the following limit?
$$\lim_{x\to 0}\frac{\left(\sin x\right)^2-\left(\sin \left(1-x\right)\right)^2}{x^2\left(\sin x\right)^2}=\lim_{x\to 0}\frac{2-2\sin x-\sin x}{x^2\left(\sin x\right)^2}=\lim_{x\to 0}\frac{2-2\sin x}{x^2}=\frac{2-2\cdot 0}{0\cdot 0}=2$$

A:

By expansion one has
$$
\sin(x)=x-\frac{x^3}3+\cdots
$$
Also by expanding the function and using that
$$
\sin(x)=x-\frac{x^3}{30}+\cdots
$$
one can see that
$$
\left(\sin(x)\right)^2=x^2-\frac{x^4}3+\cdots
$$
From this you obtain
$$
2-2\sin(x)=x^2-\frac{x^4}3+\cdots
$$
and
$$
\frac{2-2\sin(x)}{x^2}=1-\frac{x^2}3+\cdots
$$
Finally
$$
\lim_{x\to0}\frac{\left(\sin x\right)^2-\left(\sin \left(1-x\right)\right)^2}{x^2\left(\sin x\right)^2}=


E. 10.9 MB

E. 10.9 MB

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