Evaluate the Limit

Problem

(lim_x→−9)((x2−81)/(x+9))

Solution

1. Identify the type of limit by substituting x=−9 into the expression.

2. Observe that direct substitution results in the indeterminate form 0/0 because (−9)2−81=0 and −9+9=0

3. Factor the numerator using the difference of squares formula, a2−b2=(a−b)*(a+b)

x2−81=(x−9)*(x+9)

4. Simplify the expression by canceling the common factor (x+9) from the numerator and the denominator, noting that x≠−9 when evaluating the limit.

((x−9)*(x+9))/(x+9)=x−9

5. Evaluate the limit by substituting x=−9 into the simplified expression.

−9−9=−18

Final Answer

(lim_x→−9)((x2−81)/(x+9))=−18

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