We’re going to return to the definition of the Limit of a Function, and try to add some more precision to the way that we talk about these limits. We’ll recreate the definition, without some of the ambiguous terminology, and then explore the new definition a bit, in order to get a good feel for what this means.
By the end of this exploration, we should have a pretty good understanding of what a limit is, it should match our intuition that we’ve built, and we should see a path that we could walk to investigate how these limit properties and results that we’ve talked about could be described.
Let’s consider the function \(f(x)=\sqrt{x}\text{.}\) We’re going to mostly be interested in the \(x\)-values around 9, since we’ll be investigating \(\displaystyle\lim_{x\to 9}\sqrt{x}\text{.}\) We know that the \(y\)-value outputs of \(f(x)\) are going to be around 3, giving us:
Now we’re going to perform some tasks that will hopefully help us add some precision to our definition of a limit. You can use the graph and inputs below while you work.
We want to come up with points on the graph of \(f(x)\) whose \(y\)-values are within the interval \((2,4)\text{.}\) Can you find an interval of \(x\)-values around \(x=9\) where the function outputs \(f(x)\) are all in the interval \((2,4)\text{?}\)
Can you describe this in terms of distance? How far can \(x\)-values be from \(x=9\) where the corresponding \(f(x)\) values are all in \((2,4)\text{?}\)
“If I construct an interval of \(x\)-values that are within units of \(x=9\text{,}\) then the corresponding function outputs will all be within \((2,4)\text{.}\)”
Repeat this process, but for the interval \((2.9, 3.1)\text{.}\) Can you find an interval of \(x\)-values around \(x=9\) where the function outputs \(f(x)\) are all in the interval \((2.9,3.1)\text{?}\)
Can you describe this in terms of distance? How far can \(x\)-values be from \(x=9\) where the corresponding \(f(x)\) values are all in \((2.9,3.1)\text{?}\)
“If I construct an interval of \(x\)-values that are within units of \(x=9\text{,}\) then the corresponding function outputs will all be within 0.1 units of \(y=3\text{.}\)”
Come up with an even smaller interval. Change the distance between the function outputs and \(y=3\) to something smaller than 0.1. Construct a new target interval of \(y\)-values, and see if you can find an interval of \(x\)-values such that all of the function outputs are in your target interval.
Have you convinced yourself that you can always do this? Do you think it really matters what the target interval of \(y\)-values is? Can you always find an interval of \(x\)-values around \(x=9\) (or \(x\)-values that are some non-zero distance from \(x=9\)) where the outputs are in within that target distance from \(y=3\text{?}\)
Let’s formalize this: If \(\varepsilon\) is some real number with \(\varepsilon\gt 0\text{,}\) then can you find an interval of \(x\)-values around \(x=9\) where the function outputs \(f(x)\) are all in the interval \(\left( 3-\varepsilon , 3+\varepsilon \right)\text{?}\)
Can you describe this in terms of distance? How far can \(x\)-values be from \(x=9\) where the corresponding \(f(x)\) values are all in \(\left( 3-\varepsilon , 3+\varepsilon \right)\text{?}\)
“If I construct an interval of \(x\)-values that are within units of \(x=9\text{,}\) then the corresponding function outputs will all be within \(\varepsilon\) units of \(y=3\text{.}\)”
We’ll try this again, but with a new function. This will change a bit of how we think about constructing our intervals. Consider the function \(g(x)=\dfrac{x}{4}+3\text{.}\) We’re going to mostly be interested in the \(x\)-values around -2, since we’ll be investigating:
We want to come up with points on the graph of \(g(x)\) whose \(y\)-values are within the interval \((2,3)\text{.}\) Can you find an interval of \(x\)-values around \(x=-2\) where the function outputs \(g(x)\) are all in the interval \((2,3)\text{?}\)
Can you describe this in terms of distance? How far can \(x\)-values be from \(x=-2\) where the corresponding \(g(x)\) values are all in \((2,3)\text{?}\)
“If I construct an interval of \(x\)-values that are within units of \(x=-2\text{,}\) then the corresponding function outputs will all be within \((2,3)\text{.}\)”
Note that another way of describing the interval, \((2,3)\) is to say that the function outputs are all within \(\dfrac{1}{2}\) unit of \(y=\dfrac{5}{2}\text{.}\)
Repeat this process, but for the interval \(\left(\dfrac{19}{8}, \dfrac{21}{8}\right)\text{.}\) Can you find an interval of \(x\)-values around \(x=-2\) where the function outputs \(g(x)\) are all in the interval \(\left(\dfrac{19}{8}, \dfrac{21}{8}\right)\text{?}\)
Can you describe this in terms of distance? How far can \(x\)-values be from \(x=-2\) where the corresponding \(g(x)\) values are all in \(\left(\dfrac{19}{8}, \dfrac{21}{8}\right)\text{?}\)
“If I construct an interval of \(x\)-values that are within units of \(x=-2\text{,}\) then the corresponding function outputs will all be within \(\frac{1}{8}\) unit of \(y=-2\text{.}\)”
If \(\varepsilon\) is any real number with \(\varepsilon\gt 0\text{,}\) then can you find an interval of \(x\)-values around \(x=-2\) where the function outputs \(g(x)\) are all in the interval \(\left(\dfrac{5}{2}-\varepsilon, \dfrac{5}{2}+\varepsilon\right)\text{?}\)
Can you describe this in terms of distance? How far can \(x\)-values be from \(x=-2\) where the corresponding \(f(x)\) values are all in \(\left(\dfrac{5}{2}-\varepsilon, \dfrac{5}{2}+\varepsilon\right)\text{?}\)
We built a system to talk about \(y\)-values that are “arbitrarily close the single, real number, \(L\text{.}\)” We constructed an interval, \((L-\varepsilon, L+\varepsilon)\) for any \(\varepsilon\gt 0\text{.}\)
We found a way to talk about \(x\)-values being “sufficiently close to, but not equal to, \(a\text{.}\)” We constructed an interval of \(x\)-values around\(x=a\text{,}\) and then considered the function outputs from those \(x\)-values. We should clarify that we really are considering the function outputs from all of these \(x\)-values in the interval except\(x=a\text{.}\)
We tried to come up with a way of describing this interval of \(x\)-values—whose function outputs (except \(f(a)\)) are all guaranteed to be in \((L-\varepsilon, L+\varepsilon)\)—using a kind of function of \(\varepsilon\text{.}\) That way, we can know that we can repeat this process for any\(\varepsilon\) measure of “closeness.”
When we refer to that function of \(\varepsilon\) to give us the distance for the \(x\)-values, we often use the symbol, \(\delta\text{.}\) Sometimes people call this \(\delta_{\varepsilon}\) as a way of reminding ourselves that it depends on the value of \(\varepsilon\text{.}\)
Definition1.7.1.(Open Interval) Limit of a Function.
Suppose that \(f(x)\) is a function that exists for all \(x\)-values in some open interval around and containing \(a\text{,}\) except possibly at \(x=a\) itself. We say that
if, for any open interval \((p,q)\) of \(y\)-values containing the single, real number \(L\text{,}\) there is some corresponding open interval \((c, d)\) around \(x=a\) (that is, \(c\lt a\lt d\)) such that for all \(x\)-values in \((c,d)\) with \(x\neq a\text{,}\) the function output \(f(x)\) is in \((p,q)\text{.}\)
Definition1.7.2.(Epsilon-Delta) Limit of a Function.
Suppose that \(f(x)\) is a function that exists for all \(x\)-values in some open interval around and containing \(a\text{,}\) except possibly at \(x=a\) itself. We say that
if \(L\) is some single, real number, and for any real number \(\varepsilon \gt 0\text{,}\) there is a corresponding \(\delta_\varepsilon \gt0\) such that \(|f(x)-L|\lt \varepsilon\) whenever \(0\lt|x-a|\lt \delta_\varepsilon\text{.}\)
Both of these definitions are very important to the further study of calculus, but especially important it to use these definitions of limits to think about continuity.
We can say that a function is continuous at some point if every open interval in the range of the function around the \(y\)-value of the point comes from an open interval in the domain of the function around the \(x\)-value of the point. This is a very useful and important definition for the study of topology, and a nice way of generalizing the concept of continuity.
Consider the \(\varepsilon-\delta\) definition of a limit. What does \(\varepsilon\) represent? Connect it with the standard definition of a limit that we’ve been using. Similarly, what does \(\delta\) represent? Connect it with the standard definition of a limit we’ve been using.
For the function \(f(x) = -3x+1\text{,}\) explain how to find a value for \(\delta\) that corresponds with any \(\varepsilon\) so that when \(0\lt |x-2|\lt \delta\) we have \(|f(x) +5|\lt \varepsilon\text{.}\)
Consider the open interval definition of a limit. What do \((c, d)\) and \((p,q)\) represent. Connect them to the standard definition of a limit that we’ve been using.
For the function \(f(x)=x^2+4\text{,}\) explain how to find an interval \((c,d)\) around \(x=2\) if we are given some interval \((p,q)\) around \(L=8\) such that if \(x\) is in \((c,d)\) then \(f(x)\) is in \((p,q)\text{.}\)
