Evaluate the Limit

Problem

(lim_x→1)((√(,x)−x2)/(1−√(,x)))

Solution

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

2. Factor the numerator by taking out the common term √(,x)

(√(,x)*(1−x√(,x)))/(1−√(,x))

3. Rewrite the term x√(,x) as (√(,x))3 to recognize a difference of cubes in the numerator.

(√(,x)*(1−(√(,x))3))/(1−√(,x))

4. Apply the formula for the difference of cubes, a3−b3=(a−b)*(a2+a*b+b2) where a=1 and b=√(,x)

1−(√(,x))3=(1−√(,x))*(1+√(,x)+x)

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

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

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

(lim_x→1)(√(,x))*(1+√(,x)+x)

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

√(,1)*(1+√(,1)+1)=1*(1+1+1)=3

Final Answer

(lim_x→1)((√(,x)−x2)/(1−√(,x)))=3

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