Evaluate the Limit

Problem

(lim_x→−2)((x3+8)/(x2+8*x+12))

Solution

1. Identify the indeterminate form by substituting x=−2 into the expression.

((−2)3+8)/((−2)2+8*(−2)+12)=(−8+8)/(4−16+12)=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. Factor the denominator by finding two numbers that multiply to 12 and add to 8

x2+8*x+12=(x+2)*(x+6)

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

((x+2)*(x2−2*x+4))/((x+2)*(x+6))=(x2−2*x+4)/(x+6)

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

((−2)2−2*(−2)+4)/(−2+6)=(4+4+4)/4=12/4=3

Final Answer

(lim_x→−2)((x3+8)/(x2+8*x+12))=3

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