Evaluate the Limit limit as x approaches 2 of (|x-2|)/(x-2)

Problem

(lim_x→2)(|x−2|/(x−2))

Solution

1. Identify the nature of the limit by checking the behavior of the absolute value function |x−2| near x=2

2. Evaluate the left-hand limit by considering values where x<2 which implies x−2<0 and |x−2|=−(x−2)

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

3. Evaluate the right-hand limit by considering values where x>2 which implies x−2>0 and |x−2|=x−2

(lim_x→2)((x−2)/(x−2))=(lim_x→2)(1)=1

4. Compare the one-sided limits to determine if the general limit exists.

−1≠1

5. Conclude that since the left-hand limit and the right-hand limit are not equal, the limit does not exist.

Final Answer

(lim_x→2)(|x−2|/(x−2))=Does Not Exist

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