Evaluate the Limit

Problem

(lim_h→0)((√(,9+h)−3)/h)

Solution

1. Identify the indeterminate form by substituting h=0 into the expression, which results in (√(,9)−3)/0=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is √(,9+h)+3

3. Expand the numerator using the difference of squares formula (a−b)*(a+b)=a2−b2

((√(,9+h)−3)*(√(,9+h)+3))/(h*(√(,9+h)+3))=((9+h)−9)/(h*(√(,9+h)+3))

4. Simplify the numerator by subtracting the constants.

h/(h*(√(,9+h)+3))

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

1/(√(,9+h)+3)

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

1/(√(,9+0)+3)=1/(3+3)=1/6

Final Answer

(lim_h→0)((√(,9+h)−3)/h)=1/6

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