Evaluate the Limit

Problem

(lim_x→−2)((x3−4*x)/(x+2))

Solution

1. Identify the indeterminate form by substituting x=−2 into the expression, which results in ((−2)3−4*(−2))/(−2+2)=0/0

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

(x*(x2−4))/(x+2)

3. Apply the difference of squares formula a2−b2=(a−b)*(a+b) to the term x2−4

(x*(x−2)*(x+2))/(x+2)

4. Simplify the expression by canceling the common factor (x+2) from the numerator and the denominator.

x*(x−2)

5. Evaluate the limit by substituting x=−2 into the simplified expression.

−2*(−2−2)

6. Calculate the final numerical value.

−2*(−4)=8

Final Answer

(lim_x→−2)((x3−4*x)/(x+2))=8

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