Describe the long run behavior of $$ f(n)=2(n+1)(n+5)^{3}(n-5)^{2} $$ As $$ n \rightarrow-\infty, f(n) \rightarrow $$ ? © As $$ n \rightarrow \infty, f(n) \rightarrow $$ ?

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To analyze the long-run behavior of the function $$ f(n) = 2(n+1)(n+5)^{3}(n-5)^{2} $$, we can consider the leading term of the polynomial as $$ n $$ approaches both $$ -\infty $$ and $$ \infty $$.

1. **As $$ n \rightarrow -\infty $$**:
   - The function $$ f(n) $$ is a polynomial, and the leading term will dominate the behavior of the function for large absolute values of $$ n $$.
   - The leading term can be found by expanding $$ f(n) $$:
     $$
     f(n) = 2(n+1)(n+5)^{3}(n-5)^{2}
     $$
   - The leading term comes from multiplying the highest degree terms in each factor:
     - From $$ (n+1) $$, the leading term is $$ n $$.
     - From $$ (n+5)^3 $$, the leading term is $$ n^3 $$.
     - From $$ (n-5)^2 $$, the leading term is $$ n^2 $$.
   - Therefore, the leading term of $$ f(n) $$ is:
     $$
     2 \cdot n \cdot n^3 \cdot n^2 = 2n^6
     $$
   - As $$ n \rightarrow -\infty $$, $$ 2n^6 \rightarrow -\infty $$.

2. **As $$ n \rightarrow \infty $$**:
   - Similarly, as $$ n $$ approaches $$ \infty $$, the leading term $$ 2n^6 $$ will dominate.
   - Thus, as $$ n \rightarrow \infty $$, $$ 2n^6 \rightarrow \infty $$.

In summary:
- As $$ n \rightarrow -\infty, f(n) \rightarrow -\infty $$.
- As $$ n \rightarrow \infty, f(n) \rightarrow \infty $$.
As $$ n \rightarrow -\infty $$, $$ f(n) \rightarrow -\infty $$, and as $$ n \rightarrow \infty $$, $$ f(n) \rightarrow \infty $$.

Describe the long run behavior of \( f(n)=2(n+1)(n+5)^{3}(n-5)^{2} \) As \( n \rightarrow-\infty, f(n) \rightarrow \) ? © As \( n \rightarrow \infty, f(n) \rightarrow \) ?

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