Evaluate the Limit

Problem

(lim_x→10)(|x−10|/(x−10))

Solution

1. Identify the nature of the limit by checking the left-hand and right-hand limits, as the absolute value function |x−10| changes behavior at x=10

2. Evaluate the right-hand limit as x approaches 10 from the positive side (x>10, where |x−10|=x−10

(lim_x→10)((x−10)/(x−10))=(lim_x→10)(1)=1

3. Evaluate the left-hand limit as x approaches 10 from the negative side (x<10, where |x−10|=−(x−10)

(lim_x→10)((−(x−10))/(x−10))=(lim_x→10)(−)*1=−1

4. Compare the one-sided limits to determine if the general limit exists.

1≠−1

5. Conclude that because the left-hand limit and the right-hand limit are not equal, the limit does not exist.

Final Answer

(lim_x→10)(|x−10|/(x−10))=Does Not Exist

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