Evaluate the Limit ( limit as x approaches 0 of |x|)/x

Problem

(lim_x→0)(|x|/x)

Solution

1. Analyze the absolute value function |x| which is defined as x if x≥0 and −x if x<0

2. Evaluate the right-hand limit as x approaches 0 from the positive side (x>0.

(lim_x→0)(|x|/x)=(lim_x→0)(x/x)

(lim_x→0)(1)=1

3. Evaluate the left-hand limit as x approaches 0 from the negative side (x<0.

(lim_x→0)(|x|/x)=(lim_x→0)((−x)/x)

(lim_x→0)(−)*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→0)(|x|/x)=Does Not Exist

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