Evaluate the Limit

Problem

(lim_x→−2)((x+2)/(x3+8))

Solution

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

(−2+2)/((−2)3+8)=0/0

Since this is an indeterminate form, the expression must be simplified.

2. Factor the denominator 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 of x+2 from the numerator and the denominator.

(x+2)/((x+2)*(x2−2*x+4))=1/(x2−2*x+4)

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

1/((−2)2−2*(−2)+4)=1/(4+4+4)

1/(4+4+4)=1/12

Final Answer

(lim_x→−2)((x+2)/(x3+8))=1/12

Want more problems? Check here!