Evaluate the Limit

Problem

(lim_x→3)((x^3−27)/(x−3))

Solution

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

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

x^3−27=(x−3)*(x^2+3*x+9)

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

((x−3)*(x^2+3*x+9))/(x−3)=x^2+3*x+9

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

3^2+3*(3)+9=9+9+9=27

Final Answer

(lim_x→3)((x^3−27)/(x−3))=27

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