1. ## Series and sequences

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

2. Originally Posted by racewithferrari
Show that $f(x) = (x^3 + 1)/(x - 7)$ is order $x^2$
I'm sorry. What does order mean?

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

4. 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?

5. 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

$\chi$ $\sigma$