Evaluate the Limit

Problem

(lim_x→2)(|x−2|/(x−2))

Solution

1. Identify the type of limit. Since the expression involves an absolute value |x−2| the behavior of the function changes depending on whether x approaches 2 from the left or the right.

2. Evaluate the left-hand limit. For x<2 the expression inside the absolute value is negative, so |x−2|=−(x−2)

(lim_x→2)((−(x−2))/(x−2))

(lim_x→2)(−)*1=−1

3. Evaluate the right-hand limit. For x>2 the expression inside the absolute value is positive, 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 overall limit does not exist.

−1≠1

Final Answer

(lim_x→2)(|x−2|/(x−2))=Does Not Exist

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