Evaluate the Limit

Problem

(lim_x→1)((x^3−1)/(x−1))

Solution

1. Identify the indeterminate form by substituting x=1 into the expression, which results in (1^3−1)/(1−1)=0/0

2. Factor the numerator using the difference of cubes formula, a^3−b^3=(a−b)*(a^2+a*b+b^2) where a=x and b=1

x^3−1=(x−1)*(x^2+x+1)

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

((x−1)*(x^2+x+1))/(x−1)=x^2+x+1

4. Evaluate the limit by substituting x=1 into the remaining polynomial.

1^2+1+1=3

Final Answer

(lim_x→1)((x^3−1)/(x−1))=3

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