Evaluate the Limit

Problem

(lim_x→9)((x2−81)/(x−9))

Solution

1. Identify the indeterminate form by substituting x=9 into the expression, which results in 0/0

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

x2−81=(x−9)*(x+9)

3. Simplify the expression by canceling the common factor (x−9) from the numerator and the denominator.

((x−9)*(x+9))/(x−9)=x+9

4. 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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