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. Factor the denominator using the difference of squares pattern, noting that x−1=(√(,x))2−1

3. Rewrite the expression using the factored form of the denominator.

(√(,x)−1)/((√(,x)−1)*(√(,x)+1))

4. Simplify the fraction by canceling the common factor (√(,x)−1) from the numerator and the denominator.

1/(√(,x)+1)

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

1/(√(,1)+1)

Final Answer

(lim_x→1)((√(,x)−1)/(x−1))=1/2

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