Find the limit of the following sequence or determine that the sequence diverges. $$ \left\{b_{n} ight\} $$ if $$ b_{n}=\left\{\begin{array}{ll}\frac{n}{n+6} & ext { for } n \leq 4600 \ n e^{-n} & ext { for } n4600\end{array} ight. $$ Select the correct choice below and fill in any answer boxes to complete the choice. A. The limit of the sequence is $$ \square $$. (Type an exact answer.) B. The sequence diverges.

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Find the limit of the following sequence or determine that the sequence diverges. $$ \left\{b_{n}ight\} $$ if $$ b_{n}=\left\{\begin{array}{ll}\frac{n}{n+6} & 	ext { for } n \leq 4600 \ n e^{-n} & 	ext { for } n>4600\end{array}ight. $$ Select the correct choice below and fill in any answer boxes to complete the choice. A. The limit of the sequence is $$ \square $$. (Type an exact answer.) B. The sequence diverges.

UpStudy AI · Accepted Answer

1. For $$ n \leq 4600 $$, the sequence is defined as
   $$
   b_{n}=\frac{n}{n+6}.
   $$
   If one were to consider the limit of this expression as $$ n \to \infty $$, we would have
   $$
   \lim_{n\to\infty}\frac{n}{n+6} = 1.
   $$
   However, this part of the definition is only valid for $$ n \leq 4600 $$.

2. For $$ n > 4600 $$, the sequence takes the form
   $$
   b_{n}=n e^{-n}.
   $$
   To find the limit as $$ n \to \infty $$ for this piece, observe that $$ e^{-n} $$ decays exponentially fast. Even though $$ n $$ increases without bound, the exponential decay dominates the linear growth of $$ n $$. More rigorously, we can write
   $$
   \lim_{n\to\infty} n e^{-n} = \lim_{n\to\infty} \frac{n}{e^n} = 0.
   $$

3. Since the behavior of a sequence for very large $$ n $$ determines the limit, and for $$ n > 4600 $$ the expression $$ n e^{-n} $$ tends to $$ 0 $$ as $$ n \to \infty $$, the limit of the entire sequence is determined by this part.

A. The limit of the sequence is $$ 0 $$.
The limit of the sequence is $$ 0 $$.

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