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Graphical Limits

In our previous two posts, Functions and Domains, we looked at how to determine if a mapping was a function and, if it was, what its domain was. While functions our an important part of Calculus, their study really falls into earlier courses in mathematics. As such, today’s post looks at the first major topic of Calculus, the limit. We begin by looking at the concept of a limit.

Limits

The idea of a limit of a function is that we want to know what happens to the function values as the \(x\)-values get close to, but not equal to some number. That is, if we follow the \(y\)-values of a given function, we look at what they will get close to. As an example suppose that we have the following graph.

With this graph, put your finger on the graph at the point \((1,1.5)\). If you then trace the graph to your right, you will continue along this path to the point \((2,1)\). Therefore, the \(y\)-values are approaching \(1\) as the \(x\)-values approach \(2\) from the left, so we would say
\begin{align*}
\lim_{x \to 2^{-}}f(x)=1.
\end{align*}

In the same way, if you start at the point \((4,1)\) and trace the graph to your left, you will also move to the point \((2,1)\). Again, the \(y\)-values will approach \(1\) as the \(x\)-values approach \(2\) from the right. We then would say
\begin{align*}
\lim_{x \to 2^{+}}f(x)=1.
\end{align*}

Since, in both cases, the function moved to the point \((2,1)\), we would say that the \(y\)-values are approaching 1 as the \(x\)-values approach \(0\). That is,
\begin{align*}
\lim_{x \to 2}f(x)=1.
\end{align*}

Another example

Let’s now look at what happens as \(x\) approaches 0. That is, we want to find \(\lim_{x \to 0}f(x)\).

In order to find this, we begin by choosing an \(x\) values less than \(0\), say \(x=-1\). We would then place our finger on the graph and follow the graph heading right. We would then stop just before we get to the point when \(x=0\). Notice here that we would approach the point \((0,0)\), so the \(y\)-values would approach \(0\) as the \(x\)-values approach \(0\) from the left. Even though the function at 0 is \(f(0)=1\) (since this is the filled in dot) this does not affect the limit value. Remember the limit is what happens as we get close to, but not equal to, the \(x\)-value. We, therefore, have that
\begin{align*}
\lim_{x \to 0^{-}}f(x)=0.
\end{align*}

Now, if we scroll back to our graph again and start at some \(x\)-value bigger than \(0\), say \(x=.5\), we will again place our finger on the graph. Now follow the graph with your finger to the left until you are almost at the point \(x=0\). This time we will see that the function values tend to \(1\) as \(x\) gets close to \(0\). Therefore,
\begin{align*}
\lim_{x \to 0^{+}}f(x)=1.
\end{align*}

Notice that we did not get the same answer from both directions. In this case we would say that the function does not approach a single \(y\)-value as \(x\) approaches \(0\). Therefore,
\begin{align*}
\lim_{x \to 0}f(x) \text{ does not exist.}
\end{align*}

Conclusion

Hopefully you learned something today and enjoyed the post. If you did make sure to share the post with anyone else that may find it useful on Social Media. If you would like more practice with this problem, try to find \(\lim\limits_{x \to 1}f(x)\) or \(\lim\limits_{x \to -1}f(x)\). Also, if you would like a more precise and detailed definition of limit, please visit the post Proving Limits.

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