<h2>Limits</h2>
Consider the function:
\(y=\cfrac{{x}^{2}+4x-12}{x+6}\)
The numerator of the function can be factorised as:
\(y=\cfrac{(x+6)(x-2)}{x+6}.\)
Then we can cancel the \(x+6\) from numerator and denominator and we are left with:
\(y=x-2.\)
However, we are only able to cancel the \(x+6\) term if \(x\ne -6\). If \(x=-6\), then the denominator becomes \(\text{0}\) and the function is not defined. This means that the domain of the function does not include \(x=-6\). But we can examine what happens to the values for \(y\) as \(x\) gets closer to \(-\text{6}\). The list of values shows that as \(x\) gets closer to \(-\text{6}\), \(y\) gets closer and closer to \(-\text{8}\).
<table>
<tbody>
<tr>
<td>
\(x\)
</td>
<td>
\(y=\cfrac{(x+6)(x-2)}{x+6}\)
</td>
</tr>
<tr>
<td>
\(-\text{9}\)
</td>
<td>
\(-\text{11}\)
</td>
</tr>
<tr>
<td>
\(-\text{8}\)
</td>
<td>
\(-\text{10}\)
</td>
</tr>
<tr>
<td>
\(-\text{7}\)
</td>
<td>
\(-\text{9}\)
</td>
</tr>
<tr>
<td>
\(-\text{6.5}\)
</td>
<td>
\(-\text{8.5}\)
</td>
</tr>
<tr>
<td>
\(-\text{6.4}\)
</td>
<td>
\(-\text{8.4}\)
</td>
</tr>
<tr>
<td>
\(-\text{6.3}\)
</td>
<td>
\(-\text{8.3}\)
</td>
</tr>
<tr>
<td>
\(-\text{6.2}\)
</td>
<td>
\(-\text{8.2}\)
</td>
</tr>
<tr>
<td>
\(-\text{6.1}\)
</td>
<td>
\(-\text{8.1}\)
</td>
</tr>
<tr>
<td>
\(-\text{6.09}\)
</td>
<td>
\(-\text{8.09}\)
</td>
</tr>
<tr>
<td>
\(-\text{6.08}\)
</td>
<td>
\(-\text{8.08}\)
</td>
</tr>
<tr>
<td>
 \(-\text{6.01}\) 
</td>
<td>
 \(-\text{8.01}\) 
</td>
</tr>
<tr>
<td>
\(-\text{5.9}\)
</td>
<td>
\(-\text{7.9}\)
</td>
</tr>
<tr>
<td>
\(-\text{5.8}\)
</td>
<td>
\(-\text{7.8}\)
</td>
</tr>
<tr>
<td>
\(-\text{5.7}\)
</td>
<td>
\(-\text{7.7}\)
</td>
</tr>
<tr>
<td>
\(-\text{5.6}\)
</td>
<td>
\(-\text{7.6}\)
</td>
</tr>
<tr>
<td>
\(-\text{5.5}\)
</td>
<td>
\(-\text{7.5}\)
</td>
</tr>
<tr>
<td>
\(-\text{5}\)
</td>
<td>
\(-\text{7}\)
</td>
</tr>
<tr>
<td>
\(-\text{4}\)
</td>
<td>
\(-\text{6}\)
</td>
</tr>
<tr>
<td>
\(-\text{3}\)
</td>
<td>
\(-\text{5}\)
</td>
</tr>
</tbody>
</table>
The graph of this function is shown below. The graph is a straight line with slope \(\text{1}\) and \(y\)-intercept \(-\text{2}\), but with a hole at \(x=-6\). As \(x\) approaches \(-\text{6}\) from the left, the \(y\)-value approaches \(-\text{8}\) and as \(x\) approaches \(-\text{6}\) from the right, the \(y\)-value approaches \(-\text{8}\). Since the function approaches the same \(y\)-value from the left and from the right, the limit exists.
<img alt="31cf9f41815e03acaa02b1b91fbc43e1.png" src="https://data.ulearngo.com/assets/074343cc-e3c6-4d05-bb30-b6b5969069c0"/>

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Limits

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Limits

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