# inverse function , asymptotics ..

let be a function given by $f^{-1} (x) = \sqrt x + G(x)$

with $G(x)= \sum_{n=0}^{N}a_{n} \sin(nx+ \pi/4)$ finite fourier series with N big

my questio is , if for $x \rightarrow \infty$ the function can be asymptotically defined by $f(x) \sim x^{2}$

on condition that for every 'x' $G(x) \le \le \sqrt x$ so the most importan term is the SMOOTH term defined by the square root of 'x'

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Well, $G(x)$ is obviously bounded (for example by $\sum_{n=0}^N |a_n|$, there might be a tigther one), so for large $x$, the relative error will go to zero. Not the absolute error, though.
yo mean $e_{relative}= e_{real}-e_{error}/e_{real}$ and $e_{absolute}= e_{real}-e_{error}$ ?? thanks for your answer fgp –  Jose Garcia Oct 9 '12 at 20:26