Evaluate the Limit

Problem

(lim_x→1)((x2−x)/(x2−1))

Solution

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

2. Factor the numerator by taking out the greatest common factor x

x2−x=x*(x−1)

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

x2−1=(x−1)*(x+1)

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

(x*(x−1))/((x−1)*(x+1))=x/(x+1)

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

1/(1+1)=1/2

Final Answer

(lim_x→1)((x2−x)/(x2−1))=1/2

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