Evaluate the Limit limit as x approaches 2 of (x-2)/(x^3-8)

Problem

(lim_x→2)((x−2)/(x3−8))

Solution

1. Identify the indeterminate form by substituting x=2 into the expression, which results in 0/0

2. Factor the denominator 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)/((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)+4)=1/(4+4+4)

5. Calculate the final numerical value.

1/12

Final Answer

(lim_x→2)((x−2)/(x3−8))=1/12

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