Evaluate the Limit

Problem

(lim_h→0)((4*(x+h−3)2−4*(x−3)2)/h)

Solution

1. Identify the expression as the definition of the derivative for the function ƒ(x)=4*(x−3)2

2. Expand the term (x+h−3)2 by treating (x−3) as a single unit to simplify the algebra.

(x−3+h)2=(x−3)2+2*h*(x−3)+h2

3. Substitute the expansion back into the limit expression.

(lim_h→0)((4*((x−3)2+2*h*(x−3)+h2)−4*(x−3)2)/h)

4. Distribute the constant 4 across the terms in the parentheses.

(lim_h→0)((4*(x−3)2+8*h*(x−3)+4*h2−4*(x−3)2)/h)

5. Simplify the numerator by canceling the terms 4*(x−3)2 and −4*(x−3)2

(lim_h→0)((8*h*(x−3)+4*h2)/h)

6. Factor out h from the numerator to prepare for division.

(lim_h→0)((h*(8*(x−3)+4*h))/h)

7. Divide the numerator and denominator by h

(lim_h→0)(8*(x−3)+4*h)

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

8*(x−3)+4*(0)=8*x−24

Final Answer

(lim_h→0)((4*(x+h−3)2−4*(x−3)2)/h)=8*x−24

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