Evaluate the Limit

Problem

(lim_x→2)((x3−8)/(x−2))

Solution

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

(2−8)/(2−2)=0/0

2. Factor the numerator using the difference 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 simplified expression.

2+2*(2)+4=4+4+4=12

Final Answer

(lim_x→2)((x3−8)/(x−2))=12

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