Evaluate the Limit

Problem

(lim_h→0)((√(,16+h)−4)/h)

Solution

1. Identify the indeterminate form by substituting h=0 into the expression, which results in (√(,16)−4)/0=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is √(,16+h)+4

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

(√(,16+h)−4)*(√(,16+h)+4)=(16+h)−16

4. Simplify the numerator by subtracting the constants.

(16+h)−16=h

5. Rewrite the limit expression with the rationalized form.

(lim_h→0)(h/(h*(√(,16+h)+4)))

6. Cancel the common factor of h from the numerator and the denominator.

(lim_h→0)(1/(√(,16+h)+4))

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

1/(√(,16+0)+4)=1/(4+4)

Final Answer

(lim_h→0)((√(,16+h)−4)/h)=1/8

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