Final answer:
The limit of the function 2x² |x-1| as x approaches -1 is 0. This is found by evaluating the behavior of the absolute value function as x nears -1 and simplifying the expression.
Step-by-step explanation:
The student has asked to find the limit of the function 2x² |x-1| as x approaches -1. Since we are dealing with an absolute value function, we must consider the behavior of the function as x approaches -1 from both the left and the right. When x is less than 1, the expression |x-1| is equal to -(x-1), and when x is greater than or equal to 1, |x-1| is equal to x-1.
As x approaches -1, the expression simplifies to 2x²(-x + 1). Plugging -1 into this expression yields 2(-1)²(-(-1) + 1) which simplifies to 2(1)(0), giving us a limit of 0. Therefore, the limit as x approaches -1 of 2x² |x-1| is 0.