Limits of a Function

Definition of the Limit of a Function

So when you decide on the definition of the limit, from the previous examples you can see that the function value at \(x=a\) shouldn't matter. Indeed, using \(0<|x-a|<\delta\) excludes the point \(x=a\) so the limit is properly defined even if the function has a hole in it.

The notation for the limit of a function is generally

\[\lim\limits_{x\to a}f(x)=L.\]

This is read as "the limit of the function as \(x\) approaches \(a\) is equal to \(L\)".

Next let's look at the steps of applying the definition of the limit for a specific function using a graph and table.

Graphing to Find the Limit of a Function

For this example take the function to be

\[f(x)=\frac{1}{4}(x+1)(x-1)(x-5),\]

and find the limit as \(x\to 3\).

Step 1. Make a proposal for what the limit of the function should be. To do this, graph the function and make a table of the function values near \(x=3\).

Table 1. Data of the function \(\frac{1}{4}(x+1)(x-1)(x-5)\) around the point \(x=3\).

Fig. 4. Graph and Table of a Function near \(x=3\).

Make a proposal for the limit as \(x\rightarrow 3\). By looking at the table and graph above, you can see that the function value at \(x=3\) is \(-4\), and that as long as the \(x\) values are close to \(3\), the function values seem to be close to \(-4\). So \(L=-4\) is a good candidate for the limit.

Step 2. Someone gives you a value of \(\epsilon\), and you need to find the corresponding \(\delta\) value. To do that, you will need to find the range of function values that are close to your proposed limit \(L\). Since you only care about \(x\) close to \(3\) and function values close to \(L\), draw in the dotted lines corresponding to \(y=L+\epsilon\) and \(y=L-\epsilon\).

Table 2. Data of the function \(\frac{1}{4}(x+1)(x-1)(x-5)\) around the point \(x=3\).

Fig. 5. Graph and Table of a Function near \(x=3\), larger size.

Step 3. Now you can see that you really want to zoom in on the graph to get a better idea of what is happening. It can also help to add a column to your table showing how close you are to the proposed limit.

Table 3. Data of the function \(\frac{1}{4}(x+1)(x-1)(x-5)\) arund the point \(x=3\) in a smaller interval.

Fig. 6. Zoom in on limit graph and table of function values.

Looking at the table of values, you can see that \(|f(x)-L|\) is smaller when you are to the right of \(a=3\) than if you are to the left of \(a=3\). So you can think of having two different values of "how close" you need to be to \(a=3\) to be in the \(\epsilon\) window. Call those two different values \(\delta_1\) and \(\delta_2\), and graph those lines as well.

Fig. 6. Zoom in on limit graph and table of function values.

Fig. 7. Graph of function showing epsilon and delta windows.

Step 4. You have two different candidates for what \(\delta\) should be, what do you do? Since you want both

\[0<|x-3|<\delta_1\quad \text {and} \quad 0<|x-3|<\delta_2.\]

For this to be true, take \(\delta\) to be smaller than both \(\delta_1\) and \(\delta_2\). This will be the \(\delta\) you need to finish proving that

\[\lim\limits_{x\rightarrow 3} f(x)=-4.\]

Examples of the Limit of a Function

Let's look at some examples of applying the definition to find the limit of a function.

Generally you won't be able to take \(\delta\) to be anything you like! It usually depends on the function, the value of \(a\), and the \(\epsilon\) in question.

So as you can see, the value of \(\delta\) in the previous example did depend on \(a=7, f(7)\) and \(\epsilon.\)