# Examples of Non-Existing Limits

Limit of a function does not necessarily exists. Possible cases of non-existing limits would be when

1. at least one of the one-sided limits does not exist;
2. both one-sided limits exist but they are not the same.

Here are a couple of examples of non-existing limits.

Example. Let $f(x)$ be the function defined by $f(x)=\sin\frac{1}{x}$ for $x\ne 0$. The graph of this function is given by

.

As $x$ approaches to $0$, $\sin\frac{1}{x}$ keeps oscillating near the $y$-axis but it does not approach to anywhere. This is the case when neither $\lim_{x\to 0-}\sin\frac{1}{x}$ nor $\lim_{x\to 0+}\sin\frac{1}{x}$ exists. The following picture shows you a closer look at the graph near the $y$-axis.

The function $f(x)=\sin\frac{1}{x}$ is called topologist’s sine curve.

Example. Let $f(x)$ be the function defined by $f(x)=\left\{\begin{array}{ccc}x-1 & {\rm if} & x<2\$x-2)^2+3 & {\rm if} & x\geq 2.\end{array}\right.$ The graph of \(f(x)$ is

Let us calculate the left-hand and the right-hand limit of $f(x)$ at $x=2$: \begin{eqnarray*}\lim_{x\to 2-}f(x)&=&\lim_{x\to 2-}(x-1)\\&=&1,\\\lim_{x\to 2+}f(x)&=&\lim_{x\to 2+}(x-2)^2+3\\&=&3.\end{eqnarray*} Both the left-hand and the right-hand limits of $f(x)$ exist, however they do not coincide. Hence the limit $\lim_{x\to 2}f(x)$ does not exist.

## 9 thoughts on “Examples of Non-Existing Limits”

1. JKeyes

From the above graph, f(x), I am not understanding where the other plotted numbers are coming from being that x can be any number that is less than 2 or greather than or equal to 2 for the appropriate equation. However, I do understand the y-intercept of the equations can be used as a plotted point. Also, I am asking how can one identify which numbers are the limit?

1. lee Post author

Jalisa, I am not sure if I understood your question correctly. I guess that you are not sure how we obtain those values for the left-hand limit and the right-hand limit. They are obtained from the graph, and in fact we have done a similar example in class. For instance, to find the left-hand limit $\displaystyle\lim_{x\to 2-}f(x)$, first take a point $p$ on the $x$-axis that is less than 2 but close to 2. Plot the corresponding $f(p)$ on the $y$-axis. Next take another point $q$ on the $x$-axis that is less than 2 but closer to 2 than the previously chosen point $p$. Plot again the corresponding $f(q)$ on the $y$-axis. If you keep on doing this process, you will clearly see that those plotted values on the $y$-axis are getting close to 1. That way we find that $\displaystyle\lim_{x\to 2-}f(x)=1$. The right-hand limit is found in a similar manner. Hope this helps.

1. lee Post author

I think you are mistaken. $f(5)$ is not defined. You are seeing an open circle on the graph at $x=5$. That means the function $f(x)$ is not defined at $x=5$. However, the limit $\lim_{x\to 5}f(x)$ exists and the value is 4 from the graph.

2. DayJ

I’m working on the 1.3 homework assignment and I’m not understanding part of number 3. I am able to find the value of each quantity, but I can not when it does not exist nor can I explain why it does not exists.

lim f(x)
x–>1

the book says this does not exist and I don’t understand why.

1. lee Post author

From the graph, we see that the left-hand limit $\lim_{x\to 1-}f(x)=2$ while the right hand-limit $\lim_{x\to 1+}f(x)=3$. Since their values do not coincide, $\lim_{x\to 1}f(x)$ does not exist.