Limits of Functions
How function evaluations change when the argument approaches a certain point.
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Study Limits of Functions
Now we begin to introduce the concept of continuity.
Definition 1 ((One-sided) limits of Functions). Let \(I\subset\mathbb{K}\), let \(f:I\to\mathbb{K}\) be a function, and let \(x_0\in I\). Then we define
the limit of \(f\) as \(x\) tends to \(x_0\) by \(c\in\mathbb{K}\) if for all sequences \((x_n)_{n\in\mathbb{N}}\) in \(I\backslash\{x_0\}\) with \(\lim_{n\to\infty}x_n=x_0\) holds \(\lim_{n\to\infty}f(x_n)=c\). In this case, we write \[\lim_{x\to x_0}f(x)=c\]
the limit of \(f\) as \(x\) tends from the left to \(x_0\) by \(c\in\mathbb{K}\) if \(I\subset \mathbb{R}\) and if for all sequences \((x_n)_{n\in\mathbb{N}}\) in \(\{x\in I\,:\,x< x_0\}\) with \(\lim_{n\to\infty}x_n=x_0\) holds \(\lim_{n\to\infty}f(x_n)=c\). In this case, we write \[\lim_{x\nearrow x_0}f(x)=c.\]
the limit of \(f\) as \(x\) tends from the right to \(x_0\) by \(c\in\mathbb{K}\) if \(I\subset \mathbb{R}\) and if for all sequences \((x_n)_{n\in\mathbb{N}}\) in \(\{x\in I\,:\,x>x_0\}\) with \(\lim_{n\to\infty}x_n=x_0\) holds \(\lim_{n\to\infty}f(x_n)=c\). In this case, we write \[\lim_{x\searrow x_0}f(x)=c.\]
In all three cases we assume that at least one sequence \((x_n)_{n\in\mathbb{N}}\) with the stated property exists.
Remark 2. From the above definition, we can also conclude that \(\lim_{x\to x_0}f(x)\) exists in the case \(I\subset \mathbb{R}\) if and only if \(\lim_{x\nearrow x_0}f(x)\) and \(\lim_{x\searrow x_0}f(x)\) exist and are equal. In this case, there holds \[\lim_{x\nearrow x_0}f(x)=\lim_{x\searrow x_0}f(x)=\lim_{x\to x_0}f(x).\] Though not explicitly introduced in the above definition, it should be intuitively clear what is meant by the following expressions \[\lim_{x\to\infty}f(x)=y,\qquad \lim_{x\to-\infty}f(x)=y,\qquad \lim_{x\to x_0}f(x)=\infty,\qquad \lim_{x\to x_0}f(x)=-\infty.\]
Example 3.
Consider the Heaviside function \(H:\mathbb{R}\to\mathbb{R}\) with \[H(x)=\begin{cases}1&,\text{ if }x\geq 0,\\0&,\text{ if }x<0.\end{cases}\] Then we have \(\lim_{x\nearrow 0}H(x)=0\), since for all \(x_n\in\mathbb{R}\) with \(x_n<0\) holds \(H(x_n)=0\). Further, \(\lim_{x\searrow 0}H(x)=1\), since for all \(x_n\in\mathbb{R}\) with \(x_n>0\) holds \(H(x_n)=1\). The limit \(\lim_{x\to 0}H(x)\) does not exist. E.g., take the sequence \(x_n=\frac{(-1)^n}n\). Then \[H(x_n)=\begin{cases}1:&\text{if $n$ is even,}\\0:&\text{if $n$ is odd.}\end{cases}\] Hence, \((H(x_n))_{n\in\mathbb{N}}\) is divergent.
Consider the function \(f:\mathbb{R}\to\mathbb{R}\) with \[f(x)=\begin{cases}1&,\text{ if }x= 0,\\0&,\text{ if }x\neq0.\end{cases}\] Then for all sequences \((x_n)_{n\in\mathbb{N}}\) in \(\mathbb{R}\backslash\{0\}\) holds that \((f(x_n))_{n\in\mathbb{N}}\) is a constant zero sequence. Hence, \(\lim_{x\to 0}f(x)=0\).
Consider a polynomial \(p:\mathbb{R}\to\mathbb{R}\) with \(p(x)=a_nx^n+\ldots+a_1x+a_0\) for some given \(a_0,\ldots,a_n\in\mathbb{R}\). Let \(x_0\in\mathbb{R}\). By the theorem on the calculation rules for limits of sequences of real numbers, we have that for all real sequences \((x_n)_{n\in\mathbb{N}}\) converging to \(x_0\) holds that \(p(x_n)\) converges to \(p(x_0)\), i.e., \[\lim_{x\to x_0}p(x)=p(x_0).\]
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