Evaluate the Limit

Problem

(lim_h→0)(((2+h)^3−8)/h)

Solution

1. Expand the cubic expression (2+h)^3 using the binomial expansion formula (a+b)^3=a^3+3*a^2*b+3*a*b^2+b^3

(2+h)^3=2^3+3*(2^2)*(h)+3*(2)*(h^2)+h^3

(2+h)^3=8+12*h+6*h^2+h^3

2. Substitute the expanded expression back into the limit numerator.

(lim_h→0)((8+12*h+6*h^2+h^3−8)/h)

3. Simplify the numerator by subtracting the constants.

(lim_h→0)((12*h+6*h^2+h^3)/h)

4. Factor out h from the numerator to prepare for cancellation.

(lim_h→0)((h*(12+6*h+h^2))/h)

5. Cancel the common factor of h from the numerator and denominator.

(lim_h→0)(12+6*h+h^2)

6. Evaluate the limit by substituting h=0 into the remaining expression.

12+6*(0)+(0)^2=12

Final Answer

(lim_h→0)(((2+h)^3−8)/h)=12

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