Evaluate the Limit limit as x approaches 0 of ((2+x)^3-8)/x

Problem

(lim_x→0)(((2+x)3−8)/x)

Solution

1. Expand the cubic expression (2+x)3 using the binomial expansion formula (a+b)3=a3+3*a2*b+3*a*b2+b3

(2+x)3=2+3*(2)*(x)+3*(2)*(x2)+x3

(2+x)3=8+12*x+6*x2+x3

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

(8+12*x+6*x2+x3−8)/x

3. Simplify the numerator by subtracting the constants.

(12*x+6*x2+x3)/x

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

(x*(12+6*x+x2))/x

5. Divide the numerator and denominator by x noting that x≠0 as it approaches the limit.

12+6*x+x2

6. Evaluate the limit by substituting x=0 into the simplified expression.

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

Final Answer

(lim_x→0)(((2+x)3−8)/x)=12

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