Evaluate the Limit

Problem

(lim_h→0)(((4+h)2−16)/h)

Solution

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

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

2. Substitute the expanded expression back into the limit.

(lim_h→0)((16+8*h+h2−16)/h)

3. Simplify the numerator by subtracting the constant terms.

(lim_h→0)((8*h+h2)/h)

4. Factor out the common term h from the numerator.

(lim_h→0)((h*(8+h))/h)

5. Divide the numerator and denominator by h to remove the indeterminate form.

(lim_h→0)(8+h)

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

8+0=8

Final Answer

(lim_h→0)(((4+h)2−16)/h)=8

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