Evaluate the Limit

Problem

(lim_x→1)((x^3−1)/(x^4−1))

Solution

1. Identify the indeterminate form by substituting x=1 into the expression, which results in (1^3−1)/(1^4−1)=0/0

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

x^3−1=(x−1)*(x^2+x+1)

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

x^4−1=(x^2−1)*(x^2+1)

x^4−1=(x−1)*(x+1)*(x^2+1)

4. 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+1)*(x^2+1))=(x^2+x+1)/((x+1)*(x^2+1))

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

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

3/((2)*(2))=3/4

Final Answer

(lim_x→1)((x^3−1)/(x^4−1))=3/4

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