Evaluate the Limit

Problem

(lim_h→0)(((2+h)2−4)/h)

Solution

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

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

2. Substitute the expanded expression back into the limit.

(lim_h→0)((4+4*h+h2−4)/h)

3. Simplify the numerator by subtracting the constants.

(lim_h→0)((4*h+h2)/h)

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

(lim_h→0)((h*(4+h))/h)

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

(lim_h→0)(4+h)

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

4+0=4

Final Answer

(lim_h→0)(((2+h)2−4)/h)=4

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