Evaluate the Limit limit as x approaches 1 of (x-1)/(x^2-1)

Problem

(lim_x→1)((x−1)/(x2−1))

Solution

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

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

x2−1=(x−1)*(x+1)

3. Simplify the expression by canceling the common factor (x−1) from the numerator and the denominator, provided x≠1

(x−1)/((x−1)*(x+1))=1/(x+1)

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

1/(1+1)=1/2

Final Answer

(lim_x→1)((x−1)/(x2−1))=1/2

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