Final answer:
The limit of the expression as x approaches -6 does not exist because the left-hand limit and the right-hand limit are different due to the presence of the absolute value.
Step-by-step explanation:
The student asks to find the limit of the expression 6 - |x| / 6x as x approaches -6. To determine whether this limit exists, we must examine the behavior of the function from both sides of x = -6. Because of the absolute value in the numerator, we have to evaluate the limit for the two cases where x is less than -6 and where x is greater than -6.
For x < -6, |x| = -x, and the expression becomes (6 + x) / 6x. For x > -6, |x| = x, and the expression simplifies to (6 - x) / 6x. By evaluating these two cases separately, we find that the limits are different. Therefore, the limit does not exist (DNE) since the left-hand limit does not equal the right-hand limit as x approaches -6. Remember that for a limit to exist