Evaluate the Limit limit as x approaches 1 of (|x-1|)/(x-1)

Problem

(lim_x→1)(|x−1|/(x−1))

Solution

1. Analyze the absolute value expression |x−1| by considering the definition of absolute value based on the sign of the input.

2. Evaluate the left-hand limit as x approaches 1 from the negative side (x<1, where x−1 is negative and |x−1|=−(x−1)

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

3. Evaluate the right-hand limit as x approaches 1 from the positive side (x>1, where x−1 is positive and |x−1|=x−1

(lim_x→1)((x−1)/(x−1))=(lim_x→1)(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→1)(|x−1|/(x−1))=Does Not Exist

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