Question

lmt x-->0 ((xcosx-sinx)/x²sinx)​

Answer 1

Answer:

-1/3

Step-by-step explanation:

lim x-> 0 ((xcosx - sinx)/x2sinx) when we substitute 0 we get an expression in 0/0 form .

now ,

lim x->0 ((xcosx - sinx)/x2 (sinx/x)*x )     [ we are writing sin x as sin x/ x * x ]

now , sinx /x =1

so ,

lim x->0 ((xcosx - sinx)/ x3)

using L's hospital rule ,

lim x-> 0 ( cosx + x(-sinx) - (cosx)) / 3x2 )

lim x->0 (-x sinx / 3x2)

lim x->0 (-1/3(sinx/x))

as we have seen before sinx/x=1

so

lim x->0 (-1/3(1))

so , ultimately we get

lim x->0 ((xcosx - sinx) /x2sinx) = -1/3

hope this helps.