Evaluate the Limit

Problem

(lim_x→3)((x3−27)/(x2−9))

Solution

1. Identify the indeterminate form by substituting x=3 into the expression, which results in (3−27)/(3−9)=0/0

2. Factor the numerator using the difference of cubes formula, a3−b3=(a−b)*(a2+a*b+b2) where a=x and b=3

x3−27=(x−3)*(x2+3*x+9)

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

x2−9=(x−3)*(x+3)

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

((x−3)*(x2+3*x+9))/((x−3)*(x+3))=(x2+3*x+9)/(x+3)

5. Evaluate the limit by substituting x=3 into the simplified expression.

(3+3*(3)+9)/(3+3)=(9+9+9)/6

6. Reduce the resulting fraction to its simplest form.

27/6=9/2

Final Answer

(lim_x→3)((x3−27)/(x2−9))=9/2

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