Evaluate the Limit

Problem

(lim_h→0)((√(,25+h)−5)/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 expression √(,25+h)+5

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

(√(,25+h)−5)*(√(,25+h)+5)=(25+h)−25

4. Simplify the numerator by subtracting the constants.

(25+h)−25=h

5. Divide the common factor of h from the numerator and the denominator.

h/(h*(√(,25+h)+5))=1/(√(,25+h)+5)

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

1/(√(,25+0)+5)=1/(5+5)

Final Answer

(lim_h→0)((√(,25+h)−5)/h)=1/10

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