Evaluate the Limit

Problem

(lim_x→3)(ln(x2−9))

Solution

1. Identify the type of limit by substituting the value x=3 into the expression inside the natural logarithm.

2. Evaluate the inner expression as x approaches 3

(lim_x→3)(x2−9)=3−9=0

3. Analyze the behavior of the natural logarithm function ln(u) as its argument u approaches 0 from the right.

(lim_u→0)(ln(u))=−∞

4. Determine the direction of the limit. Since x2−9 must be positive for the logarithm to be defined, we consider the limit as x approaches 3 from the right (x→3.

5. Conclude that as the argument approaches 0 the natural logarithm decreases without bound.

Final Answer

(lim_x→3)(ln(x2−9))=−∞

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