Binomial distribution to Gaussian disstribution ?

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In some articles , Stirling formula is often used to approximate [tex]^{n}C{x}[/tex]
, all I can not understand is that how do you know "How many terms you need in this approximation ?"

Or , binomial distribution cannot become normal distribution forever , cos it's only an "Approximation" ?

If you have another method of proving this "Transmutation" ~ please show it to me~

Thanks ~