Evaluate the Limit

Problem

(lim_x→0)((√(,x+2)−√(,2))/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 of the numerator, which is √(,x+2)+√(,2)

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

(√(,x+2)−√(,2))*(√(,x+2)+√(,2))=(x+2)−2

4. Simplify the numerator to x and rewrite the limit expression.

(lim_x→0)(x/(x*(√(,x+2)+√(,2))))

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

(lim_x→0)(1/(√(,x+2)+√(,2)))

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

1/(√(,0+2)+√(,2))=1/(2√(,2))

7. Rationalize the denominator if necessary by multiplying the numerator and denominator by √(,2)

1/(2√(,2))⋅√(,2)/√(,2)=√(,2)/4

Final Answer

(lim_x→0)((√(,x+2)−√(,2))/x)=√(,2)/4

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