Evaluate the Limit limit as x approaches 4 of (|x-4|)/(x-4)

Problem

(lim_x→4)(|x−4|/(x−4))

Solution

1. Identify the nature of the limit by checking the behavior of the absolute value function |x−4| near x=4

2. Evaluate the left-hand limit as x approaches 4 from the negative side (x<4, where |x−4|=−(x−4)

(lim_x→4)((−(x−4))/(x−4))=(lim_x→4)(−)*1=−1

3. Evaluate the right-hand limit as x approaches 4 from the positive side (x>4, where |x−4|=x−4

(lim_x→4)((x−4)/(x−4))=(lim_x→4)(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→4)(|x−4|/(x−4))=Does Not Exist

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