<s>[INST] Why divide numerator and denominator by highest power in a rational function? $$\lim_{x \to \infty}\frac{x^2+3}{x^3+1} \;\;\;\;\;\;\;\; \textrm{or}\;\;\;\;\;\;\;\; \lim_{x \to \infty}\frac{10x^3}{2x^3+3x^2+6x}$$ 
Why do we proceed by first dividing each term in the numerator and denominator by the highest power of $x$ in the denominator?   
In the first function, we would divide each term by $x^3$, and in the second function, we would divide each term by $x^3$.  
Why divide by $x$ with the highest power in the denominator when determining the limit?  
Thank you. 
 [/INST] Because terms of the form $\frac{1}{x^n}$ converge to zero as $x \to \infty$.
In your example above, $ \lim_{x \to \infty}\frac{10x^3}{2x^3+3x^2+6x} = \lim_{x \to \infty}\frac{10}{2+3\frac{1}{x}+6\frac{1}{x^2}} = \frac{10}{2} = 5$.
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