Evaluate the Limit

Problem

(lim_x→2)((√(,x+2)−2)/(x−2))

Solution

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

4. Simplify the numerator to obtain x−2

(x−2)/((x−2)*(√(,x+2)+2))

5. Cancel the common factor of x−2 from the numerator and the denominator, provided x≠2

1/(√(,x+2)+2)

6. Evaluate the limit by substituting x=2 into the simplified expression.

1/(√(,2+2)+2)=1/(2+2)

Final Answer

(lim_x→2)((√(,x+2)−2)/(x−2))=1/4

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