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Section 3.3 Infinite limits and vertical asymptotes

So far we have studied limits that “exist”, in the sense that as \(x\) approaches \(a\text{,}\) the value of the function \(f(x)\) gets arbitrarily close to a finite value \(L\text{.}\) But there is another interesting case: sometimes the value of \(f(x)\) becomes arbitrarily large (either positive or negative) as \(x \to a\text{.}\) This is the concept of “infinite limits”, which is what we study in this section. We also study the related concept of vertical asymptotes.

Subsection 3.3.1 Instructional video

Subsection 3.3.2 Key concepts

Concept 3.3.1. Infinite limits.

Let \(f(x)\) be a function that is defined near \(x=a\) (but not necessarily at \(x=a\)).

We write

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

and say that the limit of \(f(x)\text{,}\) as \(x\) approaches \(a\text{,}\) is infinity if the values of \(f(x)\) can be made arbitrarily large and positive by taking \(x\) sufficiently close to \(a\) (on either side of \(a\)) but not equal to \(a\text{.}\)

Similarly, we write

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

and say that the limit of \(f(x)\text{,}\) as \(x\) approaches \(a\text{,}\) is negative infinity if the values of \(f(x)\) can be made arbitrarily large and negative by taking \(x\) sufficiently close to \(a\) (on either side of \(a\)) but not equal to \(a\text{.}\)

Concept 3.3.2. Vertical asymptotes.

The line \(x=a\) is called a vertical asymptote of the curve \(y=f(x)\) if either

\begin{equation*} \lim_{x \to a^-} f(x) = \pm \infty \qquad \text{or} \qquad \lim_{x \to a^+} f(x) = \pm \infty \qquad \text{or both.} \end{equation*}

Note that the function \(f(x)\) does not have to blow up on both sides of \(x=a\) for it to be a vertical asymptote; as long as the limit is infinite on one side of \(x=a\) it is a vertical asymptote.

Concept 3.3.3. Existence of limits.

We say that \(\displaystyle \lim_{x \to a} f(x)\) exists if \(\displaystyle \lim_{x \to a} f(x) = L\text{,}\) with \(L\) a finite number (\(L=0\) is perfectly fine).

So there are two reasons why a limit may not exist:

  1. The left-sided and right-sided limits may be different:
    \begin{equation*} \lim_{x \to a^+} f(x) \neq \lim_{x \to a^-} f(x), \end{equation*}
    in which case we write that
    \begin{equation*} \lim_{x \to a} f(x) \text{ DNE}. \end{equation*}
  2. The limit may be infinite. In this case we write
    \begin{equation*} \lim_{x \to a} f(x) = \pm \infty, \end{equation*}
    keeping in mind that the limit does not exist since \(\pm \infty\) is not a finite number.

Further readings 3.3.3 Further readings