Evaluate the Limit

Problem

(lim_h→0)(((3+h)(−1)−3(−1))/h)

Solution

1. Rewrite the expression by converting the negative exponents into fractions.

(1/(3+h)−1/3)/h

2. Find a common denominator for the terms in the numerator to combine them.

(3−(3+h))/(3*(3+h))/h

3. Simplify the numerator by distributing the negative sign and combining like terms.

(−h)/(3*(3+h))/h

4. Divide the fraction by h which is equivalent to multiplying by 1/h

(−h)/(3*h*(3+h))

5. Cancel the common factor of h from the numerator and the denominator, noting that h≠0

(−1)/(3*(3+h))

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

(−1)/(3*(3+0))

7. Simplify the final numerical result.

−1/9

Final Answer

(lim_h→0)(((3+h)(−1)−3(−1))/h)=−1/9

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