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 type of limit. Since the expression involves an absolute value |x−3| the behavior of the function depends on whether x approaches 3 from the left or the right.

2. Evaluate the left-hand limit as x→3 For x<3 the expression x−3 is negative, so |x−3|=−(x−3)

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

3. Evaluate the right-hand limit as x→3 For x>3 the expression x−3 is positive, so |x−3|=x−3

(lim_x→3)((x−3)/(x−3))=(lim_x→3)(1)=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→3)((x−3)/|x−3|)=Does Not Exist

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