Mathematics

The key is recognizing that for a limit to exist at a point, the left-hand and right-hand limits must be equal. For $x < 2$, simplify the rational expression by factoring—this removes the apparent discontinuity and reveals what the function actually approaches from the left. Then set that equal to what the linear piece approaches from the right, and solve for $k$.

The key is recognizing that for a limit to exist at a point, the left-hand and right-hand limits must be equal. For $x < 2$, simplify the rational expression by factoring—this removes the apparent discontinuity and reveals what the function actually approaches from the left. Then set that equal to what the linear piece approaches from the right, and solve for $k$.