Evaluate the Limit

Problem

(lim_x→−1)((x3+1)/(x+1))

Solution

1. Identify the form of the limit by substituting x=−1 into the expression.

((−1)3+1)/(−1+1)=0/0

2. Factor the numerator using the sum of cubes formula, a3+b3=(a+b)*(a2−a*b+b2)

x3+1=(x+1)*(x2−x+1)

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

((x+1)*(x2−x+1))/(x+1)=x2−x+1

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

(−1)2−(−1)+1=1+1+1=3

Final Answer

(lim_x→−1)((x3+1)/(x+1))=3

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