Evaluate the Limit

Problem

(lim_h→0)(((x+h)3−x3)/h)

Solution

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

(lim_h→0)((x3+3*x2*h+3*x*h2+h3−x3)/h)

2. Simplify the numerator by subtracting x3 from x3

(lim_h→0)((3*x2*h+3*x*h2+h3)/h)

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

(lim_h→0)((h*(3*x2+3*x*h+h2))/h)

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

(lim_h→0)(3*x2+3*x*h+h2)

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

3*x2+3*x(0)+(0)2

Final Answer

(lim_h→0)(((x+h)3−x3)/h)=3*x2

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