If `("lim")_(xvec0)(x^(-3)sin3x+a x^(-2)+b)` exists and is equal to 0, then `a=-3a n db=9/2` `a=3a n db=9/2` `a=-3a n db=-9/2` `a=3a n db=-9/2`

Options

A. `a=-3`
B. `a=0`
C. `b=1`
D. `b=-1`

Answer

A

Explanation

`underset(xto0)lim((sin3x)/(x^(3))+(a)/(x^(2))+b)=underset(xto0)lim(sin3x+ax+bx^(3))/(x^(3))`
`=underset(xto0)lim(3(sin3x)/(3x)+a+bx^(2))/(x^(2))`
For existence,
`(3+a)=0`
or `a=-3`
`:. L=underset(xto0)lim(sin3x-3x+bx^(3))/(x^(3))`
`=27underset(t to0)lim(sint-t)/(t^(3))+b=0(3x=t)`
`=-(27)/(6)+b=0`
or `b=(9)/(2)`