Evaluate the Limit limit as x approaches 1 of (x^4-1)/(x-1)

Problem

(lim_x→1)((x4−1)/(x−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, where x4−1=(x2−1)*(x2+1)

3. Factor the term (x2−1) further into (x−1)*(x+1)

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

5. Substitute the value x=1 into the remaining simplified expression (x+1)*(x2+1) to find the limit.

Final Answer

(lim_x→1)((x4−1)/(x−1))=4

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