<s>[INST] Simplifying/Converting limits I have this in my lecture:
How did 
$$\lim_{x\rightarrow \infty} x^3 \left(\tan{\frac{1}{x}}\right)\left(\sin{\frac{3}{x^2}}\right)$$ 
become 
$$\lim_{x\rightarrow \infty}3\left(\frac{\tan{\frac{1}{x}}}{\frac{1}{x}}\right)\left(\frac{\sin{\frac{3}{x^2}}}{\frac{3}{x^2}}\right)$$
Note the $x^3$ and the $3$ and denominator of the 2nd equation
UPDATE
The question is how come the $x^3$ became $3\left(\frac{1}{\frac{1}{x}}\right)\left(\frac{1}{\frac{3}{x^2}}\right)$
 [/INST] It is likely a typo.  It should probably read:
$$
3 \cdot \frac{\tan(\frac{1}{x})}{\frac{1}{x}} \cdot \frac{\sin(\frac{3}{x^2})}{\frac{3}{x^2}}
$$
which by multiplying the denominator is 
$$
\frac{3}{\frac{1}{x} \cdot \frac{3}{x^2}} \cdot \tan\left(\frac{1}{x} \right) \sin\left(\frac{3}{x^2}\right) = x^3 \tan\left(\frac{1}{x} \right) \sin\left(\frac{3}{x^2}\right).
$$
More importantly, do you see where these fractions come from?  That is, do you wee why we would want to divide $\tan(1/x)$ by $1/x$ and $\sin(3/x^2)$ by $3/x^2$?
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