Calculus 2: Examples of Non-Existing Limits

The limit of a function does not necessarily exist. Possible cases of non-existing limits may occur when

1. at least one of the one-sided limits does not exist;
2. both one-sided limits exist but they are not the same.
   

Here are a couple of examples of non-existing limits.

Example. Let \(f(x)\) be the function defined by \(f(x)=\sin\frac{1}{x}\) for \(x\ne 0\). The graph of this function is given by 

The graph of $f(x)=\sin\frac{1}{x}$

As \(x\) approaches \(0\), \(\sin\frac{1}{x}\) keeps oscillating near the \(y\)-axis and it does not approach anywhere. This is the case when neither \(\lim_{x\to 0-}\sin\frac{1}{x}\) nor \(\lim_{x\to 0+}\sin\frac{1}{x}\) exists. The following picture shows you a closer look at the graph near the \(y\)-axis.

The graph of $f(x)=\sin\frac{1}{x}$

Example. Let \(f(x)\) be the function defined by \[f(x)=\left\{\begin{array}{ccc}x-1 & {\rm if} & x<2\\(x-2)^2+3 & {\rm if} & x\geq 2.\end{array}\right.\] The graph of \(f(x)\) is

The graph of $$f(x)=\left\{\begin{array}{ccc}x-1 & {\rm if} & x<2\\(x-2)^2+3 & {\rm if} & x\geq 2.\end{array}\right.$$

Let us calculate the left-hand and the right-hand limit of \(f(x)\) at \(x=2\): \begin{align*}\lim_{x\to 2-}f(x)&=\lim_{x\to 2-}(x-1)\\&=1,\\\lim_{x\to 2+}f(x)&=\lim_{x\to 2+}(x-2)^2+3\\&=3.\end{align*} Both the left-hand and the right-hand limits of \(f(x)\) exist, however they do not coincide. Hence the limit \(\lim_{x\to 2}f(x)\) does not exist.