`L=lim_(xrarr0) (sin(sinx)-sinx)/(ax^(5)+bx^(3)+c)=-(1)/(12)`
The value/values of a is

Options

A. `in` R
B. 2
C. 0
D. 1

Answer

A

Explanation

We have `underset(xrarr0)(lim)(sin(sinx)-sinx)/(ax^(5)+bx^(3)+c)=-(1)/(12)`
`"In L.H.S. N"^(r)rarr0" when "xrarr0`
`therefore D^(t)" must "rarr0`
`rArrc=0`
Now, `underset(xrarr0)(lim)(sin(sinx)-sinx)/(ax^(5)+bx^(3))`
`=underset(xrarr0)(lim)(2sin((sinx-x)/(2))cos((sinx+x)/(2)))/(ax^(5)+bx^(3))`
`=underset(xrarr0)(lim)(2sin((sinx-x)/(2)))/(((sinx-x)/(2))).(sinx-x)/(2).(cos((sinx+x)/(2)))/(ax^(5)+bx^(3))`
`=underset(xrarr0)(lim)((sinx-x)/(ax^(5)+bx^(3)))`
`=underset(xrarr0)(lim)(cosx-1)/(5ax^(4)+3vx^(2))`
`=underset(xrarr0)(lim)(-sinx)/(20ax^(3)+6vx)" (Using L'Hospital's Rule)"`
`=underset(xrarr0)(lim)(-sinx)/(x).(1)/(20ax^(2)+6a)`
`=-(1)/(6)"for all a"inR`
`therefore" "-(1)/(6a)=-(1)/(12)rArr b=2`
`therefore" "ain R, b=2, c=0`