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 numerator using the difference of squares formula, treating x as (√(,x))2 and 1 as 1

x−1=(√(,x)−1)*(√(,x)+1)

3. Substitute the factored form back into the limit expression.

(lim_x→1)(((√(,x)−1)*(√(,x)+1))/(√(,x)−1))

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

(lim_x→1)(√(,x)+1)

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

√(,1)+1=2

Final Answer

(lim_x→1)((x−1)/(√(,x)−1))=2

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