Evaluate the Limit

Problem

(lim_x→3)(ln(x−3))

Solution

1. Identify the domain of the natural logarithm function, which requires the argument to be strictly positive: x−3>0 or x>3

2. Determine the direction of the limit, as the function is undefined for x≤3 We must evaluate the one-sided limit from the right: (lim_x→3^+)(ln(x−3))

3. Substitute a variable u=x−3 As x→3^+ the variable u→0^+

4. Evaluate the behavior of the natural log function as its argument approaches zero from the right.

(lim_u→0^+)(ln(u))=−∞

Final Answer

(lim_x→3)(ln(x−3))=−∞

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