`lim_(xrarr-7) ([x]^(2)+15[x]+56)/(sin(x+7)sin(x+8))=` (where [.] denotes the greatest integer function)

Options

A. is 0
B. is 1
C. is `-1`
D. does not exist

Answer

A

Explanation

`L=underset(xrarr-7)(lim)({x}^(2)+15[x]+56)/(sin(x+7)sin(x+8))`
`=underset(xrarr-7)(lim)(([x]+7)([x]+8))/(sin(x+7)sin(x+8))`
Now, `underset(xrarr-7^(+))(lim)(([x]+7)([x]+8))/(sin(x+7)sin(x+8))=(("exact 0").(1))/((rarr0)sin(-1))=0`
`underset(xrarr-7^(-))(lim)(([x]+7)([x]+8))/(sin(x+7)sin(x+8))=((-1).("exact 0"))/((rarr0)sin(-1))=0`
`thereforeL=0`