Short and Sweet Calculus

2.3 Infinite limits

Consider the function \(f\) defined by the equation \[f(x)=\frac{1}{(x-1)^{2}}\] The graph of this function is illustrated in Figure [fig:ch4-inf-limit-1].

Note that \(f\) is not defined at \(x=1\) (division by zero is undefined), but let’s consider the values of \(f\) when \(x\) is close to \(1\). Letting \(x\) approach \(1\) from both sides, the corresponding values of \(f\) are given in the following table.

From this table, we see that as \(x\) gets closer and closer to \(1\) but never quite equal to \(1\), \(f(x)\) becomes indefinitely large; in other words, we can make \(f(x)\) as large as we desire if we take \(x\) close enough to \(1\).

To express that \(f(x)\) increases without bound as \(x\) approaches \(1\), we write \[\lim_{x\to1}\frac{1}{(x-1)^{2}}=+\infty\] Instead of \(+\infty\), we may simply write \(\infty\).

  • The equals sign does not mean that the limit exists. Note that \(+\infty\) is not a number; it is just a symbol for indicating that \(f(x)\) indefinitely increases.

Description: Let \(f\) be a function defined on both sides of \(a\), except possibly at \(a\) itself. If we can make \(f(x)\) as large as we wish by taking \(x\) sufficiently close to \(a\) but not equal to \(a\), then we say \(f(x)\) approaches \(\infty\) as \(x\) approaches \(a\) (or the limit of \(f(x)\), as \(x\) approaches \(a\), is positive infinity and write \[\lim_{x\to a}f(x)=+\infty\]

Now consider the function \(g\) defined by the equation \[g(x)=-\frac{1}{(x-1)^{2}}.\] The graph of \(g(x)\) is shown in Figure [fig:ch4-inf-limit-4]. We notice that \(g(x)=-f(x)\) and as \(x\) approaches \(1\) from either side, \(g(x)\) decreases without bound. In this case we write \[\lim_{x\to1}g(x)=-\infty.\]

Description: Let \(g\) be a function defined on both sides of \(a\), except possibly at \(a\) itself. If we can make \(g(x)\) as large negative as we wish by taking \(x\) sufficiently close to \(a\) but not equal to \(a\), then we say \(g(x)\) approaches \(-\infty\) as \(x\) approaches \(a\) (or the limit of \(g(x)\) is as \(x\) approaches \(a\) is infinity and write \[\lim_{x\to a}g(x)=-\infty.\]

One-sided limits can be also defined accordingly. For example, consider the function \(h(x)\) defined by the equation \[h(x)=\frac{1}{x-2}.\] The graph of this function is represented in Figure [fig:Ch4-inf-limit-5]. In this case, we write \[\lim_{x\to2^{-}}h(x)=-\infty,\qquad\text{and}\qquad\lim_{x\to2^{+}}h(x)=+\infty.\]


[up][previous][table of contents][next]