Evaluate the Limit

Problem

(lim_h→0)((√(,36+h)−6)/h)

Solution

1. Identify the indeterminate form by substituting h=0 into the expression, which results in 0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is √(,36+h)+6

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

((√(,36+h)−6)*(√(,36+h)+6))/(h*(√(,36+h)+6))=((36+h)−36)/(h*(√(,36+h)+6))

4. Simplify the numerator by subtracting the constants.

h/(h*(√(,36+h)+6))

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

1/(√(,36+h)+6)

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

1/(√(,36+0)+6)=1/(6+6)=1/12

Final Answer

(lim_h→0)((√(,36+h)−6)/h)=1/12

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