Evaluate the Limit

Problem

(lim_x→−2)((x3+8)/(x2+5*x+6))

Solution

1. Substitute the value x=−2 into the expression to check for an indeterminate form.

((−2)3+8)/((−2)2+5*(−2)+6)=(−8+8)/(4−10+6)=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 6 and add to 5

x2+5*x+6=(x+2)*(x+3)

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+3))=(x2−2*x+4)/(x+3)

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

((−2)2−2*(−2)+4)/(−2+3)=(4+4+4)/1=12

Final Answer

(lim_x→−2)((x3+8)/(x2+5*x+6))=12

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