Evaluate the Limit

Problem

(lim_x→5)(|x−5|/(x−5))

Solution

1. Identify the nature of the limit. The expression involves an absolute value, which changes definition at x=5 the point where the denominator also becomes zero.

2. Evaluate the left-hand limit as x approaches 5 from the negative side (x<5. In this region, x−5 is negative, so |x−5|=−(x−5)

(lim_x→5)((−(x−5))/(x−5))=−1

3. Evaluate the right-hand limit as x approaches 5 from the positive side (x>5. In this region, x−5 is positive, so |x−5|=x−5

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

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