# 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.