Posted on Categories:Calculus Assignment, 微积分, 数学代写

# 数学代写|微积分代写Calculus代考|Limits Involving Infinity; Asymptotes of Graphs

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|微积分代写Calculus代考|Limits Involving Infinity; Asymptotes of Graphs

In this section we investigate the behavior of a function when the magnitude of the independent variable $x$ becomes increasingly large, or $x \rightarrow \pm \infty$. We further extend the concept of limit to infinite limits. Infinite limits provide useful symbols and language for describing the behavior of functions whose values become arbitrarily large in magnitude. We use these ideas to analyze the graphs of functions having horizontal or vertical asymptotes.
Finite Limits as $x \rightarrow \pm \infty$
The symbol for infinity $(\infty)$ does not represent a real number. We use $\infty$ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds.

For example, the function $f(x)=1 / x$ is defined for all $x \neq 0$ (Figure 2.49). When $x$ is positive and becomes increasingly large, $1 / x$ becomes increasingly small. When $x$ is negative and its magnitude becomes increasingly large, $1 / x$ again becomes small. We summarize these observations by saying that $f(x)=1 / x$ has limit 0 as $x \rightarrow \infty$ or $x \rightarrow-\infty$, or that 0 is a limit of $f(x)=1 / x$ at infinity and at negative infinity. Here are precise definitions for the limit of a function whose domain contains positive or negative numbers of unbounded magnitude.
DEFINITIONS

We say that $f(x)$ has the limit $L$ as $\boldsymbol{x}$ approaches infinity and write
$$\lim _{x \rightarrow \infty} f(x)=L$$
if, for every number $\varepsilon>0$, there exists a corresponding number $M$ such that for all $x$ in the domain of $f$
$$|f(x)-L|<\varepsilon \text { whenever } x>M .$$
We say that $f(x)$ has the limit $L$ as $\boldsymbol{x}$ approaches negative infinity and write
$$\lim _{x \rightarrow-\infty} f(x)=L$$
if, for every number $\varepsilon>0$, there exists a corresponding number $N$ such that for all $x$ in the domain of $f$
$$|f(x)-L|<\varepsilon \text { whenever } x<N .$$

## 数学代写|微积分代写Calculus代考|Infinite Limits

Let us look again at the function $f(x)=1 / x$. As $x \rightarrow 0^{+}$, the values of $f$ grow without bound, eventually reaching and surpassing every positive real number. That is, given any positive real number $B$, however large, the values of $f$ become larger still (Figure 2.58).
Thus, $f$ has no limit as $x \rightarrow 0^{+}$. It is nevertheless convenient to describe the behavior of $f$ by saying that $f(x)$ approaches $\infty$ as $x \rightarrow 0^{+}$. We write
$$\lim {x \rightarrow 0^{+}} f(x)=\lim {x \rightarrow 0^{+}} \frac{1}{x}=\infty$$
In writing this equation, we are not saying that the limit exists. Nor are we saying that there is a real number $\infty$, for there is no such number. Rather, this expression is just a concise way of saying that $\lim _{x \rightarrow 0^{+}}(1 / x)$ does not exist because $1 / x$ becomes arbitrarily large and positive as $x \rightarrow 0^{+}$.

As $x \rightarrow 0^{-}$, the values of $f(x)=1 / x$ become arbitrarily large and negative. Given any negative real number $-B$, the values of $f$ eventually lie below $-B$. (See Figure 2.58.) We write
$$\lim {x \rightarrow 0^{-}} f(x)=\lim {x \rightarrow 0^{-}} \frac{1}{x}=-\infty$$
Again, we are not saying that the limit exists and equals the number $-\infty$. There is no real number $-\infty$. We are describing the behavior of a function whose limit as $x \rightarrow 0^{-}$does not exist because its values become arbitrarily large and negative.

## 数学代写|微积分代写Calculus代考|Limits Involving Infinity; Asymptotes of Graphs

$$\lim {x \rightarrow \infty} f(x)=L$$ 如果，对于每个数字$\varepsilon>0$，存在一个对应的数字$M$，使得对于$f$域中的所有$x$ $$|f(x)-L|<\varepsilon \text { whenever } x>M .$$ 我们说$f(x)$有极限$L$当$\boldsymbol{x}$趋于负无穷时，我们写 $$\lim {x \rightarrow-\infty} f(x)=L$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。