Evaluate the Limit

Problem

(lim_h→0)((1/((x+h)2)−1/(x2))/h)

Solution

1. Find a common denominator for the terms in the numerator to combine the fractions.

1/((x+h)2)−1/(x2)=(x2−(x+h)2)/(x2*(x+h)2)

2. Expand the squared binomial in the numerator.

(x+h)2=x2+2*x*h+h2

3. Substitute the expansion back into the numerator and simplify.

x2−(x2+2*x*h+h2)=−2*x*h−h2

4. Rewrite the original limit expression using the simplified numerator.

(lim_h→0)((−2*x*h−h2)/(x2*(x+h)2)/h)

5. Simplify the complex fraction by dividing by h which is equivalent to multiplying the denominator by h

(lim_h→0)((−2*x*h−h2)/(h*x2*(x+h)2))

6. Factor out h from the numerator to cancel it with the h in the denominator.

(lim_h→0)((h*(−2*x−h))/(h*x2*(x+h)2))

(lim_h→0)((−2*x−h)/(x2*(x+h)2))

7. Evaluate the limit by substituting h=0 into the expression.

(−2*x−0)/(x2*(x+0)2)=(−2*x)/(x2⋅x2)

8. Simplify the final expression by canceling x from the numerator and denominator.

(−2*x)/(x4)=−2/(x3)

Final Answer

(lim_h→0)((1/((x+h)2)−1/(x2))/h)=−2/(x3)

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