Evaluate the Limit

Problem

(lim_x→3)((x2+3*x−18)/(x2−9))

Solution

1. Identify the indeterminate form by substituting x=3 into the expression, which results in (3+3*(3)−18)/(3−9)=0/0

2. Factor the numerator x2+3*x−18 into (x−3)*(x+6)

3. Factor the denominator x2−9 as a difference of squares into (x−3)*(x+3)

4. Simplify the expression by canceling the common factor (x−3) from the numerator and the denominator.

5. Evaluate the limit by substituting x=3 into the remaining expression (x+6)/(x+3)

Final Answer

(lim_x→3)((x2+3*x−18)/(x2−9))=3/2

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