1.1 Definition of a Limit

1.1.1 Functions

To begin, we pick up where Analysis 1A left off, and start talking about functions.
Definition 1.1: (Function)
Let \(A\) and \(B\) be sets. A function from \(A\) to \(B\) is a rule that assigns to each element of \(A\) a unique element of \(B\). The set \(A\) is called the domain of the function, and \(B\) is called the codomain of the function.

So, to specify a function, we need a domain, codomain and a rule that tells us how to get between the two. We can write a function \(f\) symbolically as follows: \[\begin{align*} f:\; &A \longrightarrow B,\\ &x \longmapsto f(x). \end{align*}\]

A specific type of domain we are going to need in the near future is that of a punctured neighbourhood.
Definition 1.2: (Punctured Neighbourhood)
Let \(c\in\mathbb{R}\), and \(D \subseteq \mathbb{R}\). Then \(D\) is a punctured neighbourhood of \(c\) if there exists a \(\delta_0\) such that the set \((c-\delta_0,c)\cup(c,c + \delta_0) \subseteq D.\)

For example, if we take \(c = 0\), potential punctured neighbourhoods include:

  • \(D = (-2,2)\) (This works with any \(0 < \delta_0 < 2\))
  • \(D = (-3,5)\setminus\lbrace0,1\rbrace\) (This works with any \(0 < \delta_0 < 1\))
  • \(D = \mathbb{R}\) (This works with any positive \(\delta_0\))

1.1.2 The Definition

One thing we may be interested in for a function is what happens as we approach a particular value from inside the domain. More specifically, we are interested in what happens to the corresponding codomain values: do they approach something finite, oscillate wildly, or behave even more bizarrely? In the first case, we can talk about a limiting value, of which the precise definition is formulated below:
Definition 1.3: (Limit of a Function)
Let \(c, L\in\mathbb{R}\), and let \(f: D \longrightarrow \mathbb{R}\) be a function defined on a punctured neighbourhood of \(c\). Then \(\lim_{x \to c} f(x) = L\) means \[\forall \epsilon > 0 \;\; \exists \delta > 0\;\; \text{s.t.} \;\; \forall x \in D, \;\; 0 < \lvert x - c \rvert < \delta \;\Rightarrow\; \lvert f(x) - L \rvert < \epsilon.\]

Alternatively, we can say that \(f(x) \to L\) as \(x \to c\) if this definition is satisfied. Loosely, this says that “however close you want the function to be to \(L\) at \(x = c\), you can always find a range of \(x\) values that make this possible.” Graphically, we may be in a situation as in Figure 1.1.

A diagram illustrating the definition of limits for functions.

Figure 1.1: A diagram showing the limiting process for functions. For the given \(\epsilon\) in the diagram, we see that whenever we are within a distance \(\delta\) of \(c\) in the domain, evaluating the function leaves us within a distance \(\epsilon\) of \(L\). Note that the function isn’t defined at \(c\), but the limiting value \(L\) still exists.

Once we have a definition, it makes sense to also consider its negation. Namely, we have that \(f(x)\) does not approach1 \(L\) as \(x \to c\) when \[\exists \epsilon_0 > 0\;\;\text{s.t}\;\;\forall \delta > 0,\;\; \exists x_0 \in D \;\; \text{s.t.}\;\; 0 < \lvert x_0 - c \rvert < \delta\;\; \text{and}\;\; \lvert f(x_0) - L \rvert \geq \epsilon_0.\]

1.1.3 Using the Definition in Practice

In the exam, it is highly likely that you’ll be made to use this definition to calculate a limit. In doing so, you’ll need to bound a function that is wrapped up in an absolute value. There are two results from Semester 1 that can help with this, and if you haven’t already, make sure that you commit these results to memory.
Theorem 1.1: (Triangle Inequalities)

For \(x,y \in \mathbb{R}\):

  • \(\lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert.\)
  • \(\left\lvert \lvert x \rvert - \lvert y \rvert \right\rvert \leq \lvert x - y \rvert.\)

The second result here — the reverse triangle inequality — will be incredibly useful when we need to bound fractions.

  1. This can also be written \(f(x) \not\to L\).↩︎