Evaluate the Limit

Problem

(lim_h→0)(((4+h)2−16)/h)

Solution

1. Expand the squared binomial (4+h)2 in the numerator using the formula (a+b)2=a2+2*a*b+b2

(4+h)2=16+8*h+h2

2. Substitute the expanded expression back into the limit and simplify the numerator by subtracting 16.

(16+8*h+h2−16)/h=(8*h+h2)/h

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

(h*(8+h))/h

4. Divide the numerator and denominator by h to eliminate the indeterminate form, noting that h≠0 as it approaches the limit.

8+h

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

8+0=8

Final Answer

(lim_h→0)(((4+h)2−16)/h)=8

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