Evaluate the Limit

Problem

(lim_x→3)((x3−9*x)/(x−3))

Solution

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

2. Factor the numerator by first pulling out the greatest common factor, x

(x*(x2−9))/(x−3)

3. Apply the difference of squares formula, a2−b2=(a−b)*(a+b) to the term x2−9

(x*(x−3)*(x+3))/(x−3)

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

x*(x+3)

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

3*(3+3)

3*(6)=18

Final Answer

(lim_x→3)((x3−9*x)/(x−3))=18

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