I am having trouble with this problem :
Find constants a and b so that the function is continuous for all x ∈, where
f(x)={sin(ax)/bx, x<0
2, x=0
ax+b, x>0
f(x) is continuous for $\displaystyle x\neq0$
f(x) is continuous at x = 0 if $\displaystyle \lim_{x\to 0}f(x)=f(0)$
We want $\displaystyle \lim_{x\to 0^-}\frac{sin(ax)}{bx}=2$ and $\displaystyle \lim_{x\to 0^+}ax+b=2$
$\displaystyle \lim_{x\to 0^-}\frac{sin(ax)}{bx}=\lim_{x\to 0^-}\frac{a cos(ax)}{b}=\frac{a}{b}$
$\displaystyle \frac{a}{b}=2$ so $\displaystyle a=2b$
$\displaystyle \lim_{x\to 0^+}ax+b=b$
$\displaystyle b=2$ and $\displaystyle a=4$