Evaluate the Limit

Problem

(lim_x→1)((x−1)/(√(,x)−1))

Solution

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

2. Rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is √(,x)+1

3. Expand the denominator using the difference of squares formula (a−b)*(a+b)=a2−b2 which simplifies (√(,x)−1)*(√(,x)+1) to x−1

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

5. Evaluate the remaining limit by substituting x=1 into the simplified expression √(,x)+1

Final Answer

(lim_x→1)((x−1)/(√(,x)−1))=2

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