Limits
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 \(-\text{8}\).
\(x\) | \(y=\cfrac{(x+6)(x-2)}{x+6}\) |
\(-\text{9}\) | \(-\text{11}\) |
\(-\text{8}\) | \(-\text{10}\) |
\(-\text{7}\) | \(-\text{9}\) |
\(-\text{6.5}\) | \(-\text{8.5}\) |
\(-\text{6.4}\) | \(-\text{8.4}\) |
\(-\text{6.3}\) | \(-\text{8.3}\) |
\(-\text{6.2}\) | \(-\text{8.2}\) |
\(-\text{6.1}\) | \(-\text{8.1}\) |
\(-\text{6.09}\) | \(-\text{8.09}\) |
\(-\text{6.08}\) | \(-\text{8.08}\) |
\(-\text{6.01}\) | \(-\text{8.01}\) |
\(-\text{5.9}\) | \(-\text{7.9}\) |
\(-\text{5.8}\) | \(-\text{7.8}\) |
\(-\text{5.7}\) | \(-\text{7.7}\) |
\(-\text{5.6}\) | \(-\text{7.6}\) |
\(-\text{5.5}\) | \(-\text{7.5}\) |
\(-\text{5}\) | \(-\text{7}\) |
\(-\text{4}\) | \(-\text{6}\) |
\(-\text{3}\) | \(-\text{5}\) |
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.