Evaluate the Limit

Problem

(lim_x→1)((x3−x)/(x−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 first taking out the greatest common factor x

(x*(x2−1))/(x−1)

3. Apply the difference of squares formula to factor x2−1 into (x−1)*(x+1)

(x*(x−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)

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

1*(1+1)=2

Final Answer

(lim_x→1)((x3−x)/(x−1))=2

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