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## 数学代写|微积分代写Calculus代写|Using continuity to evaluate limits

Because $f(x)=x^2+7$ is continuous at $x=3$, the two answers must be the same; the limit $\lim {x \rightarrow 3} f(x)$ (calculated on the left) must equal the value of the function $f(3)$ (calculated on the right). This means that when finding limits of continuous functions, we can try to evaluate the function first. If the function is defined, then we have found the limit. Otherwise, we must try other methods. Example 3 Find $\lim {x \rightarrow 5} \frac{x^2-1}{x+4}$
Solution Rational functions are continuous, so we start by trying to evaluate using continuity:
$$\lim {x \rightarrow 5} \frac{x^2-1}{x+4}=\frac{5^2-1}{5+4}=\frac{24}{9} .$$ Because the expression is defined at $x=5$ (the answer is a real number, $\frac{24}{9}$ ), evaluating using continuity is successful and the limit is $\frac{24}{9}$ Reading Exercise 27 Use continuity to evaluate $\lim {x \rightarrow 2} \frac{x+1}{x^2+3}$.
Example 4 Find $\lim {x \rightarrow 2} \frac{x^2-4}{x-2}$ Solution We start by trying to evaluate using continuity: $$\lim {x \rightarrow 2} \frac{x^2-4}{x-2}=\frac{2^2-4}{2-2}=\frac{0}{0},$$
which is undefined. Therefore, $x=2$ is not in the domain of the expression and we cannot evaluate the limit using continuity; we must use other methods.
Returning to our previous methods, we use $x=2+\alpha$ :
\begin{aligned} \lim _{x \rightarrow 2} \frac{x^2-4}{x-2} &=\frac{(2+\alpha)^2-4}{2+\alpha-2}=\frac{4+4 \alpha+\alpha^2-4}{\alpha} \ & \approx \frac{4 \alpha}{\alpha}=4 . \end{aligned}
The value of the limit is 4 .
From now on, our first thought when evaluating the limit of a continuous function should be to try to evaluate using continuity. When that fails, we use other methods.

## 数学代写|微积分代写Calculus代写|Intermediate value theorem

Many theorems of calculus can be understood visually, but only if enough attention is paid to learning the graphical interpretation or meaning of various terms. Among these theorems is the intermediate value theorem (IVT).

Suppose that $f$ is a continuous function defined on the closed interval $[a, b]$ and that $f(a) \neq f(b)$, as in figure 7 .Each of the functions pictured in figure 7 passes through all the $y$ coordinates between $f(a)$ and $f(b)$ at least once. If $N$ is a real number between $f(a)$ and $f(b)$, it appears that the function must take on that value at least once between $a$ and $b$. That is, if the line $y=N$ is drawn,

We can summarize our conclusion by saying that for any real number $N$ between $f(a)$ and $f(b)$, there is at least one real number $c$ between $a$ and $b$ such that $f(c)=N$. As long as $f$ is continuous, this must be the case, because one of the two points $(a, f(a))$ and $(b, f(b))$ is above the line $y=N$ and the other is below that line. If we must draw the graph in one piece between those two points, the graph must intersect the line. But, if the function is not continuous, then the conclusion might not hold, because there could be a jump in the graph so that the function’s graph jumps the line and never intersects it, as in figure 10 . Continuity is the key to the IVT.

# 微积分代考

## 数学代写|微积分代写Calculus代写|Using continuity to evaluate limits

$$\lim x \rightarrow 5 \frac{x^2-1}{x+4}=\frac{5^2-1}{5+4}=\frac{24}{9} .$$

$$\lim x \rightarrow 2 \frac{x^2-4}{x-2}=\frac{2^2-4}{2-2}=\frac{0}{0},$$

$$\lim _{x \rightarrow 2} \frac{x^2-4}{x-2}=\frac{(2+\alpha)^2-4}{2+\alpha-2}=\frac{4+4 \alpha+\alpha^2-4}{\alpha} \quad \approx \frac{4 \alpha}{\alpha}=4 .$$

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## 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。

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