Evaluate the Limit

Problem

(lim_x→−1)((x2−1)/(x+1))

Solution

1. Identify the form of the limit by substituting x=−1 into the expression.

((−1)2−1)/(−1+1)=0/0

2. Factor the numerator 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, noting that x≠−1 as we approach the limit.

((x−1)*(x+1))/(x+1)=x−1

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

−1−1=−2

Final Answer

(lim_x→−1)((x2−1)/(x+1))=−2

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