Evaluate the Limit

Problem

(lim_x→−3)((x2+6*x+9)/(x4−81))

Solution

1. Identify the indeterminate form by substituting x=−3 into the expression, which results in 0/0

2. Factor the numerator as a perfect square trinomial.

x2+6*x+9=(x+3)2

3. Factor the denominator using the difference of squares formula twice.

x4−81=(x2−9)*(x2+9)

x4−81=(x−3)*(x+3)*(x2+9)

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

((x+3)2)/((x−3)*(x+3)*(x2+9))=(x+3)/((x−3)*(x2+9))

5. Evaluate the limit by substituting x=−3 into the simplified expression.

(−3+3)/((−3−3)*((−3)2+9))=0/(−6*(18))

0/(−108)=0

Final Answer

(lim_x→−3)((x2+6*x+9)/(x4−81))=0

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