The Concept of a Limit

Section 4.1 The Concept of a Limit

Subsection 4.1.1 Limit Definitions

In Section 3.2, I said that algebra had a hard time with growth rates. The tools of algebrae don’t have the ability to calculate this thing I called a derivative. I need a new mathematical tool. That new tool is something called a limit.

Algebra can give approximate answers to a variety of questions like the question of growth rate. It can build an approximation process. Calculus starts by asking how to somehow calculate the end of the approximation processs. Is there an exact answer at the end of all the approximation? A limit is a way of understanding an infinite process and asking where the process eventually leads. The limit is the key new tool that transcends algebra and creates calculus.

The definition of limits will be restricted to limits of functions. In a limit, I want to compare the input and output of a function during an approximation process. The process starts by moving the input towards a specific value and then observing what happens to the output.

Let \(f(x)\) be a function and \(a\) a point either in the domain of the function or on the edge of its domain. Then the statement

\begin{equation*} \lim_{x \rightarrow a} f(x) = L \end{equation*}

means that as \(x\) (the input) gets closer and closer to \(a\text{,}\) f(x) (the output) gets closer and closer to \(L\text{.}\) If such an \(L\) exists, it is called limit of \(f(x)\) at \(x=a\text{.}\) I can also simply say that the limit at \(x=a\) converges. If no such \(L\) exists, the limit diverges.

Figure 4.1.1. A Convergent Limit

There are several ways in which the limit can diverge. The statement

\begin{equation*} \lim_{x \rightarrow a} f(x) = \infty \end{equation*}

means that as \(x\) gets closer and to \(a\text{,}\) the function value \(f(x)\) gets larger and larger without bound. This is a divergent limit, since \(\infty\) is not a number. \(\infty\) is just a convenient shorthard for ‘larger and larger without bound’. Similarly, the statement

\begin{equation*} \lim_{x \rightarrow a} f(x) = -\infty \end{equation*}

means that as \(x\) gets closer and to \(a\text{,}\) the function value \(f(x)\) becomes a larger and larger negative value without bound. Finally, the statement

\begin{equation*} \lim_{x \rightarrow a} f(x) \hspace{1cm} \text{ DNE } \end{equation*}

means that the limit does not exist; it doesn’t approach any number at all. DNE stands for ‘does not exist’.

Subsection 4.1.2 Vertical Asymptotes

For limits which approach \(\pm \infty\text{,}\) the graph of the function approaches a verticle line. These lines are called vertical asymptotes for the functions. Vertical asymptotes are shown as the dotted lines in the Figure 4.1.2.

Figure 4.1.2. Three Divergent Limits

Subsection 4.1.3 One-Sided Limits

In the above definitions, I assumed that \(x \rightarrow a\) means \(x\) approaches \(a\) from both sides, considering \(x\) slightly larger and \(x\) slightly smaller than \(a\text{.}\) Sometimes it is convenient to only use one side. These are called one sided limits. If I want to approach from the left (from \(x\) slightly smaller than \(a\)), I adjust the limit notation slightly by writing \(a^-\text{.}\)

\begin{equation*} \lim_{x \rightarrow a^-} f(x) \end{equation*}

If I want to approach from the right (from \(x\) slightly larger than \(a\)), I write \(a^+\text{.}\)

\begin{equation*} \lim_{x \rightarrow a^+} f(x) \end{equation*}

Note that that the notation \(a^{-1}\) is not \(-a\text{.}\) If I write \(\lim_{x \rightarrow 8^{-1}}\text{,}\) I am interested the behaviour for numbers very slightly smaller than \(8\text{:}\) for \(7.9\text{,}\) \(7.99\text{,}\) \(7.999\text{,}\) and so on. I am not at all interesting in anything near \(-8\text{.}\)

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