Series and sequences

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November 4th 2009, 05:48 PM
racewithferrari

Series and sequences

Show that $f(x) = (x^3 + 1)/(x - 7)$ is order $x^2$
November 4th 2009, 06:02 PM
Drexel28

Quote:

Originally Posted by racewithferrari

Show that $f(x) = (x^3 + 1)/(x - 7)$ is order $x^2$

I'm sorry. What does order mean?
November 4th 2009, 06:58 PM
racewithferrari

I don't know but my prof said that start by considering (x-7)f(x) and use theorems on the combination of functions.
November 4th 2009, 07:08 PM
Drexel28

Quote:

Originally Posted by racewithferrari

I don't know but my prof said that start by considering (x-7)f(x) and use theorems on the combination of functions.

What makes sense is to say that $f(x)=\mathcal{O}\left(x^2\right)$ or that $\lim_{x\to\infty}\frac{f(x)}{x^2}=c$ for some $c\in\mathbb{R}$ . Does that sound feasible?
November 5th 2009, 05:49 AM
chisigma

The 'division' between polynomials can be performed as...

$f(x)=\frac{x^{3}+1}{x-7} = \frac{x^{3}-7x^{2}}{x-7} + \frac{7x^{2} +1}{x-7}=$

$= x^{2} + \frac{7x^{2}-49x}{x-7} + \frac{49x+1}{x-7}=$

$= x^{2} +7x + \frac{49x-343}{x-7} + \frac{344}{x-7}= x^{2} + 7x + 49 + \frac{344}{x-7}$

... so that is...

$f(x)= \mathcal{O} \{x^{2}\}$

Kind regards

$\chi$ $\sigma$

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