Evaluate the Limit

Problem

(lim_x→−2)(|x+2|/(x+2))

Solution

1. Identify the type of limit. The expression involves an absolute value, which changes behavior depending on whether the argument x+2 is positive or negative.

2. Evaluate the left-hand limit as x approaches −2 from the left (x<−2. In this case, x+2<0 so |x+2|=−(x+2)

(lim_x→−2)((−(x+2))/(x+2))=(lim_x→−2)(−)*1=−1

3. Evaluate the right-hand limit as x approaches −2 from the right (x>−2. In this case, x+2>0 so |x+2|=x+2

(lim_x→−2)((x+2)/(x+2))=(lim_x→−2)(1)=1

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

−1≠1

Final Answer

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

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