Evaluate the Limit

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, a2−b2=(a−b)*(a+b)

x4−1=(x2−1)*(x2+1)

3. Factor the term (x2−1) further using the difference of squares formula.

(x2−1)*(x2+1)=(x−1)*(x+1)*(x2+1)

4. Substitute the factored form back into the limit expression.

(lim_x→1)(((x−1)*(x+1)*(x2+1))/(x−1))

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

(lim_x→1)(x+1)*(x2+1)

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

(1+1)*(1+1)=2⋅2=4

Final Answer

(lim_x→1)((x4−1)/(x−1))=4

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