<s>[INST] limits of polynomials A few polynomial limits:
$$\lim_{x\rightarrow 0} \frac{x^2 + 5x}{x^3+6x^2+3x}=\lim_{x\rightarrow 0} \frac{x + 5}{x^2+6x+3}=\frac{5}{3}$$
$$\lim_{x\rightarrow 0} \frac{x^4 + 5x}{x^3+6x^2+3x}=\lim_{x\rightarrow 0} \frac{x^3 + 5}{x^2+6x+3}=\frac{5}{3}$$
$$\lim_{x\rightarrow 0} \frac{x^4}{x^6}=\infty$$
Please verificate if I'm doing it correctly. For which two polynomials $W$, $P$ limit of  $W(x)/P(x)$ as x goes to $0$ can be a real number?
EDIT
Why for example 
$$\lim_{x\rightarrow0}\frac{x+1}{x^2}=\infty$$
$$\lim_{x\rightarrow0}\frac{x}{x^2+1}=0$$
 [/INST] Lets denote by $n$ the multiplicity of $0$ as a root of $W$ and by $k$ the multiplicity of $0$ as a root of $P$.Also denote $w(x)=\dfrac{W(x)}{x^n}$ and $p(x)=\dfrac{P(x)}{x^k}$. If $k=0$ then $\lim_{x\to0}\dfrac{W(x)}{P(x)}=\dfrac{W(0)}{P(0)}$.
If $n=k>0$ then $\lim_{x\to0}\dfrac{W(x)}{P(x)}=\dfrac{w(0)}{p(0)}$.
If $n>k>0$ then $\lim_{x\to0}\dfrac{W(x)}{P(x)}=0$.
If $n<k>0$ and $k-n$ is even then  $\lim_{x\to0}\dfrac{W(x)}{P(x)}=\text{sign}(w(0))\text{sign}(p(0))\infty$.
If $n<k>0$ and $k-n$ is odd then the $\lim_{x\to0}\dfrac{W(x)}{P(x)}$ does not exist.
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