Evaluate the Limit

Problem

(lim_h→0)((√(,14*h+1)−1)/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 √(,14*h+1)+1

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

(lim_h→0)(((√(,14*h+1)−1)*(√(,14*h+1)+1))/(h*(√(,14*h+1)+1)))

4. Simplify the numerator by squaring the terms.

(lim_h→0)(((14*h+1)−1)/(h*(√(,14*h+1)+1)))

5. Cancel the constant terms in the numerator and then divide out the common factor of h from the numerator and denominator.

(lim_h→0)((14*h)/(h*(√(,14*h+1)+1)))

(lim_h→0)(14/(√(,14*h+1)+1))

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

14/(√(,14*(0)+1)+1)

14/(1+1)

Final Answer

(lim_h→0)((√(,14*h+1)−1)/h)=7

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