Evaluate the Limit limit as x approaches 3 of (|x-3|)/(x-3)

Problem

(lim_x→3)(|x−3|/(x−3))

Solution

1. Identify the nature of the limit by checking the left-hand and right-hand limits, as the absolute value function |x−3| changes definition at x=3

2. Evaluate the right-hand limit by considering x>3 where |x−3|=x−3

(lim_x→3)((x−3)/(x−3))=(lim_x→3)(1)=1

3. Evaluate the left-hand limit by considering x<3 where |x−3|=−(x−3)

(lim_x→3)((−(x−3))/(x−3))=(lim_x→3)(−)*1=−1

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

1≠−1

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

Final Answer

(lim_x→3)(|x−3|/(x−3))=Does Not Exist

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