Evaluate the Limit

Problem

(lim_x→0)((√(,4+x)−2)/x)

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which results in (√(,4)−2)/0=0/0

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

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

(√(,4+x)−2)*(√(,4+x)+2)=(4+x)−4

4. Simplify the numerator by subtracting the constants.

(4+x)−4=x

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

x/(x*(√(,4+x)+2))=1/(√(,4+x)+2)

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

1/(√(,4+0)+2)=1/(2+2)

Final Answer

(lim_x→0)((√(,4+x)−2)/x)=1/4

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