Evaluate the Limit

Problem

(lim_x→1)((x4−1)/(x6−1))

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 squares formula a2−b2=(a−b)*(a+b)

x4−1=(x2−1)*(x2+1)

3. Factor the denominator using the difference of squares formula.

x6−1=(x3−1)*(x3+1)

4. Apply further factoring to the terms (x2−1) and (x3−1) using the difference of squares and difference of cubes formulas.

x2−1=(x−1)*(x+1)

x3−1=(x−1)*(x2+x+1)

5. Substitute the factored forms back into the limit expression.

(lim_x→1)(((x−1)*(x+1)*(x2+1))/((x−1)*(x2+x+1)*(x3+1)))

6. Cancel the common factor (x−1) from the numerator and the denominator.

(lim_x→1)(((x+1)*(x2+1))/((x2+x+1)*(x3+1)))

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

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

8. Simplify the resulting fraction.

4/6=2/3

Final Answer

(lim_x→1)((x4−1)/(x6−1))=2/3

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