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

Problem

(lim_x→0)(x/|x|)

Solution

1. Identify the nature of the function ƒ(x)=x/|x| by considering the definition of the absolute value |x|

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