Evaluate the Limit

Problem

(lim_x→2)((x^3−8)/(x^2−4))

Solution

1. Identify the indeterminate form by substituting x=2 into the expression, which results in (2^3−8)/(2^2−4)=0/0

2. Factor the numerator using the difference of cubes formula, a^3−b^3=(a−b)*(a^2+a*b+b^2) where a=x and b=2

x^3−8=(x−2)*(x^2+2*x+4)

3. Factor the denominator using the difference of squares formula, a^2−b^2=(a−b)*(a+b) where a=x and b=2

x^2−4=(x−2)*(x+2)

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

((x−2)*(x^2+2*x+4))/((x−2)*(x+2))=(x^2+2*x+4)/(x+2)

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

(2^2+2*(2)+4)/(2+2)=(4+4+4)/4=12/4=3

Final Answer

(lim_x→2)((x^3−8)/(x^2−4))=3

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