Evaluate the Limit

Problem

(lim_x→0)((√(,x+5)−√(,5))/x)

Solution

1. Identify the indeterminate form by substituting x=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 √(,x+5)+√(,5)

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

((√(,x+5)−√(,5))*(√(,x+5)+√(,5)))/(x*(√(,x+5)+√(,5)))

4. Simplify the numerator by performing the subtraction.

((x+5)−5)/(x*(√(,x+5)+√(,5)))

x/(x*(√(,x+5)+√(,5)))

5. Cancel the common factor of x from the numerator and the denominator.

1/(√(,x+5)+√(,5))

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

1/(√(,0+5)+√(,5))

1/(2√(,5))

7. Rationalize the final result by multiplying the numerator and denominator by √(,5)

√(,5)/10

Final Answer

(lim_x→0)((√(,x+5)−√(,5))/x)=√(,5)/10

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