Evaluate the Limit

Problem

(lim_x→−2)((x3+8)/(x+2))

Solution

1. Identify the indeterminate form by substituting x=−2 into the expression, which results in 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=2

x3+8=(x+2)*(x2−2*x+4)

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

((x+2)*(x2−2*x+4))/(x+2)=x2−2*x+4

4. Evaluate the limit by substituting x=−2 into the remaining quadratic expression.

(−2)2−2*(−2)+4=4+4+4=12

Final Answer

(lim_x→−2)((x3+8)/(x+2))=12

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