Evaluate the Limit

Problem

(lim_x→−1)((x2+2*x+1)/(x4−1))

Solution

1. Identify the indeterminate form by substituting x=−1 into the expression, which results in ((−1)2+2*(−1)+1)/((−1)4−1)=0/0

2. Factor the numerator as a perfect square trinomial: x2+2*x+1=(x+1)2

3. Factor the denominator using the difference of squares: x4−1=(x2−1)*(x2+1)

4. Factor the term (x2−1) further using the difference of squares: (x2−1)*(x2+1)=(x−1)*(x+1)*(x2+1)

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

((x+1)2)/((x−1)*(x+1)*(x2+1))=(x+1)/((x−1)*(x2+1))

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

(−1+1)/((−1−1)*((−1)2+1))=0/(−2*(2))=0

Final Answer

(lim_x→−1)((x2+2*x+1)/(x4−1))=0

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