Evaluate the Limit

Problem

(lim_x→−3)((x3+27)/(x+3))

Solution

1. Identify the form of the limit by substituting x=−3 into the expression, which results in the indeterminate form 0/0

2. Factor the numerator using the sum 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. Simplify the expression by canceling the common factor (x+3) from the numerator and the denominator.

((x+3)*(x2−3*x+9))/(x+3)=x2−3*x+9

4. Evaluate the limit by substituting x=−3 into the simplified polynomial expression.

(−3)2−3*(−3)+9=9+9+9

5. Calculate the final numerical value.

9+9+9=27

Final Answer

(lim_x→−3)((x3+27)/(x+3))=27

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