Find the values of $a$ and $b$ if the limits exist.

Question

Could someone solve this $$f(x) = \begin{cases} x^3+a, & x<0 \\ a\sin\frac{\pi{x}}{3}+b, & 0\leq {x}<2 \\ 3, & x=2 \\ \log_2x^{b+1}, & x>2\\ \end{cases}$$ If $\lim_{x\to0}$ and $\lim_{x\to2}$ both exist, find the value of a and b.

Answer 1

We have $$\lim_{x\rightarrow 0^-}f(x)=a; \lim_{x\rightarrow 0^+}f(x)=b;$$ $$\lim_{x\rightarrow 2^-}f(x)=b+a\frac{\sqrt{3}}{2}; \lim_{x\rightarrow 2^+}f(x)=b+1$$

Because $\exists \lim_{x\rightarrow 0}f(x)$ and $\exists \lim_{x\rightarrow 2}f(x)$, then

$a=b$ and $b+1=b+a\frac{\sqrt{3}}{2}$ so $a=b=\frac{2\sqrt{3}}{3}$