Limits

There are functions which do not have a y-value at some value of x, which means that the function does not exist on that value of x. However, in some cases the function in those x-values is still of interest, but finding the y-value by substituting the x-value will not work in these cases, so another method for finding these y-values is needed.

That is where limits come in place. Limits are used to approach the y-value by approaching the x-value that is needed to be substituted.

When a limit is written, it is always written as shown below:

Which can be read as the limit of $x$ going to $a$, in the function $f(x)$.

Limits have many uses and one of them is finding 'holes' in functions if there are any. An example of this use is shown below.

Example

In this example the function $f(x)=\frac{x^{2}-5x+6}{x-2}$ is used and the graph of this function is shown in Figure 1. We want to find the coördinates of the 'hole' in the graph.

As can be seen in Figure 1, on the point $(2,-1)$ there is a 'hole'. This 'hole' cannot be found by substituting 2 in the function, since it will result in $\frac{0}{0}$ which is undefined, so the answer will be approached by applying limits. In this case it can be easily shown that when filling in an x-value of $2$, the answer is undefined. So the limit of $x$ going to $2$ will be used here.

1. $\lim_{x\to2} \frac{x^{2}-5x+6}{x-2}$
2. $\lim_{x\to2} \frac{(x-2)(x-3)}{x-2}$ (Factorizing)
3. $\lim_{x\to2} x-3$ (Dividing the same factors away)
4. $2-3 = -1$ (Substituting the limit in the function to get the final answer)

Thus the 'hole' of the function indeed lays on the point $(2,-1)$.

Other uses

There are many other uses for limits but an easy use is shown as an example of how limits are written. These other uses include finding what a graph such as the logarithm does when the x-value goes to infinity, or finding what the tangent function does at its asymptotes. More uses will be explained in more detail in the next few articles about limits.

Conclusion

To summarize, limits can be used to find function values at places where they don't really exist. These function values are then approached by using a limit.