Limits

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.

31cf9f41815e03acaa02b1b91fbc43e1.png

[Attributions and Licenses]


This is a lesson from the tutorial, Differential Calculus and you are encouraged to log in or register, so that you can track your progress.

Log In

Share Thoughts