Evaluate the Limit

Problem

(lim_x→2)(ln(x−2))

Solution

1. Identify the domain of the natural logarithm function ln(u) which requires the argument u to be strictly greater than zero.

2. Analyze the behavior of the argument x−2 as x approaches 2 from the right side (x→2.

3. Observe that as x→2 the expression x−2 approaches 0 from the positive side.

4. Determine the limit of the natural logarithm as its argument approaches 0 from the right, noting that ln(u)→−∞ as u→0

5. Conclude that since the function approaches −∞ from the right and is undefined for x≤2 the two-sided limit does not exist as a finite number.

Final Answer

(lim_x→2)(ln(x−2))=−∞

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