If `lim_(xto1)(asin(x-1)+bcos(x-1)+4)/(x^(2)-1)=-2`, then `|a+b|` is__________.

Options

A. does not exist (in R)
B. is equal to 8
C. is equal to 15
D. is equal to 120

Answer

`(8)`

Explanation

Since RHS is finite quantity, at `xto1`, numerator must be 0. Therefore,
`0+b+4=0`
`b=-4`
Then `underset(xto1)lim(asin(x-1)-4cos(x-1)+4)/((x^(2)-1))=-2`
Put `x=1+h`. Then `underset(hto0)lim(asin h+4(1-cos h))/(h(2+h))=-2`
or `underset(hto0)lim(a((sin h)/(h))+4((1-cos h)/(h)))/(2+h)=-2`
or `" "(a(1)+0)/(2)=-2`
or `a=-4`
or `|a+b|=8`