<s>[INST] Limits and continuity: $\lim_{ x\rightarrow0} {\frac{ax+bx^3}{x^2}} = 2$. Find the values of $a$ and $b$ Please help
$$\lim_{ x\rightarrow0} {\frac{ax+bx^3}{x^2}} = 2$$
Find the values of $a$ and $b$. 
 [/INST] \begin{align}
\lim_{ x\rightarrow0} \frac{ax+bx^3}{x^2}
&=\lim_{ x\rightarrow0} \frac{x^2(\frac{a}{x}+bx)}{x^2}\\[6px]
&=\lim_{ x\rightarrow0} \left(\frac{a}{x}+bx\right)\\[6px]
&=\begin{cases}
  \pm\infty&\text{if $a\neq 0$ (*)}\\[6px]
  0 &\text{if $a=0$}
\end{cases}
\end{align}
(*) Depending on the sign of $a$ and if it is $x\to 0^+$ or $x\to 0^-$.
Then, if $a\neq 0$, the limit doesn’t exist. Therefore $a=0$, but in this case, the limit is $0$. The limit is never $2$.
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