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# 数学代写|微积分代写Calculus代考|Limits and horizontal asymptotes

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## 数学代写|微积分代写Calculus代考|Limits and horizontal asymptotes

Up till now, I’ve been looking at limits where $x$ approaches a regular, finite number. But $x$ can also approach infinity or negative infinity. Limits at infinity exist when a function has a horizontal asymptote. For example, the function in Figure 2-3 has a horizontal asymptote at $y=1$, which the function crawls along as it goes toward infinity to the right and negative infinity to the left. (Going left, the function crosses the horizontal asymptote at $x=-7$ and then gradually comes down toward the asymptote.) The limits equal the height of the horizontal asymptote and are written as
$$\lim {x \rightarrow \infty} f(x)=1 \text { and } \lim {x \rightarrow-\infty} f(x)=1$$

Instantaneous speed
Say you decide to drop a ball out of your second-story window. Here’s the formula that tells you how far the ball has dropped after a given number of seconds (ignoring air resistance):
$$h(t)=16 t^2$$
(where $h$ is the height the ball has fallen, in feet, and $t$ is the amount of time since the ball was dropped, in seconds)
If you plug 1 into $t, h$ is 16 ; so the ball falls 16 feet during the first second. During the first 2 seconds, it falls a total of $16 \cdot 2^2$, or 64 feet, and so on. Now, what if you wanted to determine the ball’s speed exactly 1 second after you dropped it? You can start by whipping out this trusty ol’ formula:
Distance $=$ rate $\cdot$ time, so Rate $=$ distance $/$ time

## 数学代写|微积分代写Calculus代考|Limits and Continuity

A continuous function is simply a function with no gaps – a function that you can draw without taking your pencil off the paper. Consider the four functions in Figure 2-5.

Whether or not a function is continuous is almost always obvious. The first two functions in Figure 2-5 $-f$ and $g$ – have no gaps, so they’re continuous. The next two $-p$ and $q-$ have gaps at $x=3$, so they’re not continuous. That’s all there is to it. Well, not quite. The two functions with gaps are not continuous everywhere, but because you can draw sections of them without taking your pencil off the paper, you can say that parts of them are continuous. And sometimes a function is continuous everywhere it’s defined. Such a function is described as being continuous over its entire domain, which means that its gap or gaps occur at $x-$ values where the function is undefined. The function $p$ is continuous over its entire domain; $q$, on the other hand, is not continuous over its entire domain because it’s not continuous at $x=3$, which is in the function’s domain.

Continuity and limits usually go hand in hand. Look at $x=3$ on the four functions in Figure 2-5. Consider whether each function is continuous there and whether a limit exists at that $x$-value. The first two, $f$ and $g$, have no gaps at $x=3$, so they’re continuous there. Both functions also have limits at $x=3$, and in both cases, the limit equals the height of the function at $x=3$, because as $x$ gets closer and closer to 3 from the left and the right, $y$ gets closer and closer to $f(3)$ and $g(3)$, respectively.

## 数学代写|微积分代写Calculus代考|Limits and horizontal asymptotes

$$\lim {x \rightarrow \infty} f(x)=1 \text { and } \lim {x \rightarrow-\infty} f(x)=1$$

$$h(t)=16 t^2$$
($h$是球下落的高度，单位是英尺，$t$是球下落的时间，单位是秒)

## 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。