Evaluate the Limit

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. Simplify the expanded expression by calculating the numerical coefficients.

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

3. Substitute the expanded form back into the original limit expression.

(lim_x→0)((8+12*x+6*x2+x3−8)/x)

4. Subtract the constants in the numerator to simplify the expression.

(lim_x→0)((12*x+6*x2+x3)/x)

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

(lim_x→0)((x*(12+6*x+x2))/x)

6. Divide the numerator and denominator by x to remove the indeterminate form.

(lim_x→0)(12+6*x+x2)

7. Evaluate the limit by substituting x=0 into the remaining polynomial.

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

Final Answer

(lim_x→0)(((2+x)3−8)/x)=12

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