Evaluate the Limit

Problem

(lim_x→1)((x3−1)/(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 using the difference of cubes formula, a3−b3=(a−b)*(a2+a*b+b2) where a=x and b=1

x3−1=(x−1)*(x2+x+1)

3. Factor the denominator using the difference of squares formula, a2−b2=(a−b)*(a+b) where a=x and b=1

x2−1=(x−1)*(x+1)

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

((x−1)*(x2+x+1))/((x−1)*(x+1))=(x2+x+1)/(x+1)

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

(1+1+1)/(1+1)=3/2

Final Answer

(lim_x→1)((x3−1)/(x2−1))=3/2

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