To determine the limit of the sequence as [tex]\( n \)[/tex] approaches infinity, we start by analyzing the given expression for [tex]\( a_n \)[/tex]:
[tex]\[
a_n = \frac{4 + n - 4n^2}{5n^2 + 9}
\][/tex]
Step 1: Identify the highest power of [tex]\( n \)[/tex] in both the numerator and the denominator. In this case, the highest power of [tex]\( n \)[/tex] in the numerator is [tex]\( n^2 \)[/tex] and in the denominator is also [tex]\( n^2 \)[/tex].
Step 2: To simplify the expression, we divide every term in the numerator and the denominator by [tex]\( n^2 \)[/tex]:
[tex]\[
a_n = \frac{\frac{4}{n^2} + \frac{n}{n^2} - 4}{5 + \frac{9}{n^2}}
\][/tex]
[tex]\[
a_n = \frac{\frac{4}{n^2} + \frac{1}{n} - 4}{5 + \frac{9}{n^2}}
\][/tex]
Step 3: As [tex]\( n \)[/tex] approaches infinity, the terms [tex]\(\frac{4}{n^2}\)[/tex], [tex]\(\frac{1}{n}\)[/tex], and [tex]\(\frac{9}{n^2}\)[/tex] all approach 0. Thus, the expression simplifies to:
[tex]\[
a_n = \frac{0 + 0 - 4}{5 + 0} = \frac{-4}{5} = -\frac{4}{5}
\][/tex]
Therefore, the limit of the sequence as [tex]\( n \)[/tex] approaches infinity is:
[tex]\[
\lim_ {n \rightarrow \infty} a_n = -\frac{4}{5}
\][/tex]
