Evaluate the Limit

Problem

(lim_x→−2)((x2+4*x+4)/(x4−16))

Solution

1. Identify the indeterminate form by substituting x=−2 into the expression.

((−2)2+4*(−2)+4)/((−2)4−16)=(4−8+4)/(16−16)=0/0

2. Factor the numerator, which is a perfect square trinomial.

x2+4*x+4=(x+2)2

3. Factor the denominator using the difference of squares formula a2−b2=(a−b)*(a+b)

x4−16=(x2−4)*(x2+4)

4. Factor the term (x2−4) further using the difference of squares.

(x2−4)*(x2+4)=(x−2)*(x+2)*(x2+4)

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

(lim_x→−2)(((x+2)2)/((x−2)*(x+2)*(x2+4)))

6. Simplify the expression by canceling the common factor (x+2) from the numerator and denominator.

(lim_x→−2)((x+2)/((x−2)*(x2+4)))

7. Evaluate the limit by substituting x=−2 into the simplified expression.

(−2+2)/((−2−2)*((−2)2+4))=0/(−4*(4+4))

8. Calculate the final numerical value.

0/(−32)=0

Final Answer

(lim_x→−2)((x2+4*x+4)/(x4−16))=0

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