Evaluate the Limit

Problem

(lim_x→1)((√(,x+3)−2)/(x−1))

Solution

1. Identify the form of the limit by substituting x=1 into the expression, which results in the indeterminate form 0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is √(,x+3)+2

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

(√(,x+3)−2)*(√(,x+3)+2)=(x+3)−4

4. Simplify the numerator to obtain x−1

5. Cancel the common factor of x−1 from both the numerator and the denominator.

(x−1)/((x−1)*(√(,x+3)+2))=1/(√(,x+3)+2)

6. Evaluate the limit by substituting x=1 into the simplified expression.

1/(√(,1+3)+2)=1/(2+2)

Final Answer

(lim_x→1)((√(,x+3)−2)/(x−1))=1/4

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