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Course Notes for Calculus II

Remkes Kooistra

Section 1.3 Epsilon-Delta Limits

Subsection 1.3.1

Our definition of limits in Calculus I was intuitive. The limit
\begin{equation*} \lim_{x \rightarrow a} f(x) = L \end{equation*}
meant as \(x\) gets closer and closer to a, \(f(x)\) gets closer and closer to \(L\text{.}\) This is intuitive, since we haven’t really said what this ‘closer and closer’ actually means. ’Closer and closer’ is only sense of movement — not a formal definition.
Some version of this limit definition have existed for two thousand years of mathematical history, from Zeno and Archimedes in ancient Greece through the early calculus into the 19th century. At some level, there were always concerns and problems: what did we actually mean by this ‘closer and closer’? A definitive answer was finally given by Weierstrass, who invented a technique now known as an epsilon-delta (\(\epsilon-\delta\)) arguments. The basic idea is to measure this ‘closer and closer’.
What does ‘\(x\) is close to \(a\)’ mean? According to Weierstass, it means there is some small positive number \(\delta > 0\) such that \(|x-a| \lt \delta\text{.}\) With this idea, we will present the definition of the limit now, formally, as an \(\epsilon-\delta\) argument.

Definition 1.3.1.

\begin{equation*} \lim_{x \rightarrow a} f(x) = L \text{ means } \forall \epsilon > 0 \ \ \exists \delta > 0 \text{ such that } |x-a| \lt \delta \implies |f(x) - L| \lt \epsilon\text{.} \end{equation*}
This is the formal limit definition. Let’s clarify what’s going on in the notation. First: what do the numbers \(\epsilon\) and \(\delta\) do? They measure the ‘closer and closer’ indicated in the intuitive limit. \(\epsilon\) is how close the function is to the limit value \(L\text{,}\) and \(\delta\) is how close the input is to the value \(a\text{.}\) They are always thought of as very small positive numbers.
Second, we have quantifiers: \(\forall\) means ‘for all’ and \(\exists\) means ‘there exists’. Moreover, the order of the quantifiers is very important. The start of the definition reads: for any (small) \(\epsilon\text{,}\) there exists a (small) \(\delta\text{.}\) That means that \(\delta\) is chosen in response to \(\epsilon\text{.}\) This creates a kind of game: the limit definition says that no matter what \(\epsilon\) you choose, in response, I can choose a \(\delta\) to make it work. \(\delta\) is dependant on \(\epsilon\) because of the order of the qualifiers.
Lastly, we have the two inequalities: these are the measure of how close we are. The order and the implication here is important: being close in \(x\) to \(a\) implies that we are close in \(f(x)\) to \(L\text{.}\) This matches the intuitive notion of the limit: closeness in \(x\) implies closeness in \(f(x)\text{.}\)
Putting it together, we get this idea: no matter how close you want to be (\(\epsilon\)) to \(L\text{,}\) we can find a distance (\(\delta\)) to \(a\) such that starting \(x\) within \(\delta\) of \(a\text{,}\) we will have \(f(x)\) within \(\epsilon\) of \(L\text{.}\)
Let’s start with a simple limit.
\begin{equation*} \lim_{x \rightarrow 2} 4x = 8 \end{equation*}
We need to know how to choose \(\delta\) in response to \(\epsilon\text{.}\) If, for example, \(\epsilon = \frac{1}{10}\) then take \(\delta = \frac{1}{40}\text{.}\) This allows a calculation of the relative closenesses.
\begin{equation*} |x-2| \lt \frac{1}{40} \implies |4x-8| = 4 |x-2| \lt 4 \frac{1}{40} = \frac{1}{10} \end{equation*}
We can generalize this calculation. If we are given an \(\epsilon\text{,}\) we can choose \(\delta = \frac{\epsilon}{4}\) such that
\begin{equation*} |x-2| \lt \delta \implies |4x-8| = 4 |x-2| \lt 4 \delta = \epsilon\text{.} \end{equation*}
This example shows the general technique: we choose \(\delta\) depending on \(\epsilon\text{,}\) and then argue that this choice of \(\delta\) gives the implication \(|x-a| \lt \delta \implies |f(x) - L| \lt \epsilon\text{.}\)
\begin{equation*} \lim_{x \rightarrow 2} x^2 = 4\text{.} \end{equation*}
This one is a bit more challenging, since the choice of \(delta\) is a little more subtle. Given \(\epsilon\text{,}\) we can choose \(\delta\) to be either \(\frac{1}{2}\) or \(\frac{\epsilon}{5}\text{,}\) whichever is smaller. Since we insist that \(\epsilon \leq \frac{1}{2}\text{,}\) \(|x-2|\lt \delta\) implies that \(x\) is close to \(2\text{,}\) so \(x+2\) is close to \(4\text{.}\) In particular, we know that \(|x+2|\lt 5\text{.}\) This gives us the two necessary pieces which we use in prove the limit implication.
\begin{equation*} |x-2| \lt \delta \implies |x^2-4| = |(x-2)(x+2)| = |x-2||x+2| \lt 5 \frac{\epsilon}{5} = \epsilon\text{.} \end{equation*}

Subsection 1.3.2 Infinite Limits

For limits which diverge to \(\infty\text{,}\) or limits as \(x \rightarrow \infty\text{,}\) we need to replace the \(\epsilon-\delta\) definition by a similar definition. With infinite limits, the intuition is no longer ‘getting closer and closer’. Instead, the intuition is that the values are ‘getting larger and larger without bound’. That is encoded by saying that for any natural number \(M\) or \(N\text{,}\) we will eventually exceed \(M\) or \(N\text{.}\) With this idea, we can encode the following limit statements formally.

Definition 1.3.4.

\begin{equation*} \lim_{x \rightarrow \infty} f(x) = L \text{ means } \forall \epsilon > 0 \exists M \in \NN \text{ such that } x > M \implies |f(x) - L| \lt \epsilon\text{.} \end{equation*}

Definition 1.3.5.

\begin{equation*} \lim_{x \rightarrow a} f(x) = \infty \text{ means } \forall N \in \NN \exists \delta > 0 \text{ such that } |x-a| \lt \delta \implies f(x) > N\text{.} \end{equation*}

Definition 1.3.6.

\begin{equation*} \lim_{x \rightarrow \infty} f(x) = \infty \text{ means } \forall N \in \NN 0 \exists M \in \NN \text{ such that } x > M \implies f(x) > N\text{.} \end{equation*}
In using the definition, we always tell how to choose the second quantity based on the first (\(M\) in terms of \(\epsilon\text{,}\) \(\delta\) in terms of \(N\) or \(M\) in terms of \(N\text{,}\) respectively). Then we use the relationship between the two terms (with some algebra with absolute values and inequalities) to prove the desired implication.
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