# Limits of Functions

How function evaluations change when the argument approaches a certain point.

## Discover Bridges #

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__Show requirements__

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Concept | Content |
---|---|

Sequences | These object are needed to define limits later on. |

Convergence | Convergent sequences have a well-defined limit. |

Limit Theorems | Combining limits is a useful tool to deduce convergence and the limit of a more complicated sequence from the convergence of simpler building blocks. |

__Show consequences__

__Show consequences__

Concept | Content |
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Continuity | The concept that relates functions with convergent sequences. |

Theorem of l'Hospital | An important tool for calculating limits. |

Improper Riemann Integrals | How to integrate on unbounded domains. |

## 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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## Solve the WeBWorK Exercise #

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