Evaluate the Limit

Problem

(lim_x→−6)((2*x+12)/|x+6|)

Solution

1. Factor the numerator by pulling out the common factor of 2

(2*(x+6))/|x+6|

2. Evaluate the left-hand limit as x approaches −6 from the negative side (x<−6. In this case, x+6 is negative, so |x+6|=−(x+6)

(lim_x→−6)((2*(x+6))/(−(x+6)))=−2

3. Evaluate the right-hand limit as x approaches −6 from the positive side (x>−6. In this case, x+6 is positive, so |x+6|=x+6

(lim_x→−6)((2*(x+6))/(x+6))=2

4. Compare the one-sided limits. Since the left-hand limit (−2 and the right-hand limit (2 are not equal, the two-sided limit does not exist.

−2≠2

Final Answer

(lim_x→−6)((2*x+12)/|x+6|)=Does Not Exist

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