Evaluate the Limit

Problem

(lim_x→3)((5*x^3−135)/(x−3))

Solution

1. Identify the indeterminate form by substituting x=3 into the expression, which results in (5*(27)−135)/(3−3)=0/0

2. Factor out the greatest common factor from the numerator.

(5*x^3−135)/(x−3)=(5*(x^3−27))/(x−3)

3. Apply the formula for the difference of cubes, 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)

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

(5*(x−3)*(x^2+3*x+9))/(x−3)=5*(x^2+3*x+9)

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

5*(3^2+3*(3)+9)=5*(9+9+9)

5*(27)=135

Final Answer

(lim_x→3)((5*x^3−135)/(x−3))=135

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