<s>[INST] Limits - a different approach $\lim_{x \to\infty }(\frac{x^3+4x^2+3x+5}{x^2+2x+3}+lx+m)=10$.
How do I calculate the value of l and m? 
My try: I know questions having limit tending to infinity can be solved by dividing the numerator and denominator by greatest power of $x$.But it got me nowhere in this question. Any help appreciated.
 [/INST] Indeed, we have
$$\lim_{x\to\infty }\left(\frac{x^3+4x^2+3x+5}{x^2+2x+3}-[-lx+10-m]\right)=0$$.
$$-l=\lim_{x\to\infty }\frac{\frac{x^3+4x^2+3x+5}{x^2+2x+3}}{x}=1$$
thus
$l=-1$
and 
$$10-m=\lim_{x\to\infty }\left(\frac{x^3+4x^2+3x+5}{x^2+2x+3}-x\right)=2$$
therefore $m=8$
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