Evaluate the Limit limit as x approaches 1 of (x-1)/(x^3-1)

Problem

(lim_x→1)((x−1)/(x^3−1))

Solution

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

2. Factor the denominator using the difference of cubes formula, a^3−b^3=(a−b)*(a^2+a*b+b^2)

3. Rewrite the expression with the factored denominator:

(x−1)/((x−1)*(x^2+x+1))

4. Cancel the common factor (x−1) from the numerator and the denominator, provided x≠1

1/(x^2+x+1)

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

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

Final Answer

(lim_x→1)((x−1)/(x^3−1))=1/3

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