Evaluate the Limit limit as x approaches 5 of (|x-5|)/(x-5)

Problem

(lim_x→5)(|x−5|/(x−5))

Solution

1. Analyze the absolute value expression |x−5| which changes behavior depending on whether x is greater than or less than 5

2. Evaluate the left-hand limit as x approaches 5 from the negative side (x<5, where |x−5|=−(x−5)

(lim_x→5)((−(x−5))/(x−5))=(lim_x→5)(−)*1=−1

3. Evaluate the right-hand limit as x approaches 5 from the positive side (x>5, where |x−5|=x−5

(lim_x→5)((x−5)/(x−5))=(lim_x→5)(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→5)(|x−5|/(x−5))=Does Not Exist

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