Reviewing Calculus, I am facing the problem:
if
$f(x)= \begin{cases} \sqrt[3]{x^2-8x^3}+ax+b,& \text{if } x\in\mathbb Q\\\\x\sin\big(\frac{1}{x}\big),& x\in \mathbb R-\mathbb Q \end{cases}$ has a limit at $ +\infty$, what would $ab$ be?
I doubted if I could treat this function as other piecewise function with some known domains (like $-7
Let the functions $f_1(x)$ and $f_2(x)$ have limits on $\mathbb R$ when $x\to +\infty$ so the function:
$f(x)= \begin{cases} f_1(x),& x\in\mathbb Q\\\\f_2(x),& x\in \mathbb R-\mathbb Q \end{cases}$ has limit at $+\infty$ if $\lim_{x\to +\infty}f_1(x)=\lim_{x\to +\infty}f_2(x)$
May I ask someone explain this hint? Thanks.