continuity

Hey, u hv 2 find d values widout using series expansion or l-hospital..............

we need to find
lim(x→o)( sin3x+A sin2x + B sinx)/x⁵
writing sin3x= 3sinx - 4sin³x and sin2x=2sin x cos x

we have (3sinx-4sin³x + 2Asinx cosx + Bsinx)/x⁵
take sinx as common
sinx( 3- 4 sin² x + 2Acosx +B)/x⁵
you can apply limit later also but i don't think it would make a difference and for my convenience i would now write lim(x→0) sinx/x=1
here only ...( you can do it afterwards if you want ,as i said it would hardly make a difference but remember you should not do this is if sinx/x is in addition with some other term )

coming back to the question
[3- 4sin²x + 2A(1-2sin²(x/2)) + B]/x⁴ so 2A + B +3=0 Otherwise this limit won't exist

now we have with us

-4[ sin²x + Asin²(x/2)]/x⁴ write sinx=2sin(x/2)cos(x/2) and take sin²(x/2) as common and again cancel with x² i.e. apply the limit

you will get

-(4+A - 4 sin²(x/2))/x²
clearly A+4=0
SO A= -4 AND B =5 AND to add up the value of the limit will be 1

please pardon any calculation mistakes (if any)