Evaluate the limit Lim (x-2)/|x-2|x→2
To evaluate the limit of (x-2)/|x-2| as x approaches 2, we need to investigate the behavior of the function as x approaches 2 from the left and from the right
To evaluate the limit of (x-2)/|x-2| as x approaches 2, we need to investigate the behavior of the function as x approaches 2 from the left and from the right.
As x approaches 2 from the left side (x < 2), the expression x-2 is negative, and |x-2| is evaluated as -(x-2). Therefore, the function becomes (x-2)/-(x-2), which simplifies to -1. As x approaches 2 from the right side (x > 2), the expression x-2 is positive, and |x-2| is evaluated as x-2. Therefore, the function becomes (x-2)/(x-2), which simplifies to 1.
Since the limit of the function as x approaches 2 from the left side is -1 and the limit as x approaches 2 from the right side is 1, we can conclude that the limit of (x-2)/|x-2| as x approaches 2 does not exist.
In other words, the function has no well-defined limit as x approaches 2.
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