\( \lim _{x \rightarrow-2}([[x]]+[[-x]])=( \) where \( [[\cdot]] \) is the greatest integer function \( ) \)

To find the limit of the expression \( \lim _{x \rightarrow-2}([[x]]+[[-x]]) \), we need to evaluate the greatest integer function for both \( x \) and \( -x \) as \( x \) approaches -2. The greatest integer function, denoted as \( [[\cdot]] \), rounds down to the nearest integer. So, for \( x = -2 \), we have: - \( [[-2]] = -2 \) (since -2 is already an integer) - \( [[-(-2)]] = [[2]] = 2 \) (since 2 is an integer) Now, we can substitute these values into the expression: \( \lim _{x \rightarrow-2}([[x]]+[[-x]]) = \lim _{x \rightarrow-2}(-2 + 2) = \lim _{x \rightarrow-2}0 = 0 \) Therefore, the limit of the expression as \( x \) approaches -2 is 0.