Evaluate the Limit

Problem

(lim_x→0)(sin(x)/|x|)

Solution

1. Identify the definition of the absolute value function |x| which behaves differently depending on whether x is positive or negative.

|x|=x* if *x>0

|x|=−x* if *x<0

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

(lim_x→0)(sin(x)/x)=1

3. Evaluate the left-hand limit as x approaches 0 from the negative side, where |x|=−x

(lim_x→0)(sin(x)/(−x))=−1⋅(lim_x→0)(sin(x)/x)=−1

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

(lim_x→0)(sin(x)/|x|)≠(lim_x→0)(sin(x)/|x|)

Final Answer

(lim_x→0)(sin(x)/|x|)=Does Not Exist

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