f(x)=x3sin3x+αsinx−βcos3x is continuous at x=0.
x→0limx33x−(33x3)+⋯+α(3x−3x3)−β(1−2(3x)2…)=f(0)
Continuing with the limit:
x→0limx3−β+x(3+α)+29βx2+(−327+3α)x3…=f(0)
For existence:
β=0,3+α=0,−327+3α=f(0)
Calculating:
α=−3,−627=−63=f(0)
f(0)=6−27+3=−4