Evaluate the limit Lim (x-2)/|x-2|x→2
To evaluate the limit Lim (x-2)/|x-2|x→2, we need to consider the limit from both the left and right sides of x=2 separately
To evaluate the limit Lim (x-2)/|x-2|x→2, we need to consider the limit from both the left and right sides of x=2 separately.
First, let’s consider the limit as x approaches 2 from the left side (x < 2): Since x < 2, we have |x-2| = 2 - x. Thus, the expression becomes (x-2)/(|x-2|) = (x-2)/(2-x). We can simplify this expression by factoring out -1 from the numerator: (x-2)/(2-x) = -(2-x)/(2-x) = -1. Therefore, when x approaches 2 from the left side, the value of the expression is -1. Now, let's consider the limit as x approaches 2 from the right side (x > 2):
Since x > 2, we have |x-2| = x – 2. The expression remains the same: (x-2)/(|x-2|) = (x-2)/(x-2).
We can simplify this expression by canceling out the common factor: (x-2)/(x-2) = 1.
Therefore, when x approaches 2 from the right side, the value of the expression is 1.
Since the limit from the left side is -1 and the limit from the right side is 1, the overall limit does not exist.
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