Evaluate the Limit

Problem

(lim_h→0)(((x+h)^3−x^3)/h)

Solution

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

(lim_h→0)((x^3+3*x^2*h+3*x*h^2+h^3−x^3)/h)

2. Simplify the numerator by subtracting x^3 from x^3

(lim_h→0)((3*x^2*h+3*x*h^2+h^3)/h)

3. Factor out the common factor h from the terms in the numerator.

(lim_h→0)((h*(3*x^2+3*x*h+h^2))/h)

4. Divide the numerator and denominator by h to eliminate the indeterminate form.

(lim_h→0)(3*x^2+3*x*h+h^2)

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

3*x^2+3*x(0)+(0)^2

Final Answer

(lim_h→0)(((x+h)^3−x^3)/h)=3*x^2

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