Evaluate the Limit

Problem

(lim_x→3)(ln(x2−9))

Solution

1. Identify the type of limit and the behavior of the inner function u=x2−9 as x approaches 3 from the right.

2. Evaluate the limit of the inner function by substituting values slightly greater than 3

(lim_x→3)(x2−9)=0

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

(lim_u→0)(ln(u))=−∞

4. Conclude that as x2−9 approaches 0 from the right, the natural log of that expression decreases without bound.

Final Answer

(lim_x→3)(ln(x2−9))=−∞

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