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    <name>
      <text>8.3.2 (Example of finding limits)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h3>Example</h3>

<p>The deer population in a forest is estimated to drop by \(7⋅3%\) each year. Each year, \(20\) deer are introduced to the forest. The initial deer population is \(200\).</p>

<ol type="a">
<li>How many deer will there be in the forest after \(3\) years?</li>
<li> What is the long term effect on the population?</li>
</ol>

<h3>Solution</h3>

<ol type="a">
<li>To answer this question, we find the values of \(u_1,u_2,u_3\) from the recurrence relation as we have seen previously. In this case we have
\[
\begin{align*}
&u_{n+1} = 0.927u_n + 20;
\\
&u_0 = 200
\\
&u_1 = 0.927 \cdot 200 + 20 =205.4
\\
&u_2 = 0.927 \cdot 205.4 + 20 =210.4058
\\
&u_3 = 0.927 \cdot 210.4058 + 20 = 215.0461.
\end{align*}
\]
Thus, there are \(215\) deer after three years.</li>
<li>In order to see what the overall long term effect on the deer population is, we have to calculate the limit of the sequence. Firstly, note that a limit exists because 
\[
-1 < 0.927 < 1.
\]
Therefore we can calculate the limit as
\[
L= \frac{b}{1-a} = \frac{20}{1-0.927} = 273.97
\]
to 2 d.p. This tells us that the deer population will settle around \(273\).
</li>
</ol>]]></text>
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