## 数学代写|微积分代写Calculus代写|Limit of a Function

Definition of a Limit:
Let $f(x)$ be defined for all $x$ in an interval about $x=a$, but not necessarily at $x=a$. If there is a number $L$ such that to each positive number $\varepsilon$ there corresponds a positive number $\delta$ such that
$$|f(x)-L|<\varepsilon \text { provided } 0<|x-a|<\delta,$$
we say that $L$ is the limit of $f(x)$ as $x$ approaches $a$ and write
$$\lim {x \rightarrow a} f(x)=L .$$ The ordinary algebraic manipulations can be performed with limits as shown in Appendix A2; thus, $$\lim {x \rightarrow a}[F(x)+G(x)]=\lim {x \rightarrow a} f(x)+\lim {x \rightarrow a} G(x) .$$
Two trigonometric limits are of particular interest (Appendix A3):
$$\lim {\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1 \text { and } \lim {\theta \rightarrow 0} \frac{1-\cos \theta}{\theta}=0 .$$
The following limit is of such great interest in calculus that it is given the special name $e$, as discussed in frame 109 and Appendix A3:
$$e=\lim _{x \rightarrow 0}(1+x)^{1 / x}=2.71828 \ldots$$

## 数学代写|微积分代写Calculus代写|Differentials (frames 264–273)

If $x$ is an independent variable and $y=f(x)$, the differential $d x$ of $x$ is defined as the increment, $x_2-x_1$, where $x_1$ is the point of interest. The differential $d x$ can be positive or negative, large or small, as we please. Then $d x$, like $x$, is an independent variable. The differential $d y$ is then defined by the following rule: $d y=\gamma^{\prime} d x$ where $\gamma^{\prime}$ is the derivative of $\gamma$ with respect to $x$. Although the meaning of the derivative, $\gamma^{\prime}$, is $\lim {\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$, we see that this can also be interpreted as the ratio of the differentials $d y$ and $d x$. As discussed in frames 265 and 266 , $d y$ is not the same as $\Delta \gamma$, although \begin{aligned} &\lim {d x=\Delta x \rightarrow 0} \frac{d y}{\Delta y}=1 . \ &\lim _{\Delta x \rightarrow 0} \Delta y=d y \end{aligned}
Differentiation formulas can easily be written in terms of differentials. Thus if $y=x^n$,
$$d y=d\left(x^n\right)=\left(\frac{d}{d x} x^n\right) d x=n x^{n-1} d x .$$
The relation, $\frac{d x}{d y}=\frac{1}{d y / d x}$, implied by the differential notation can be extremely useful. It’s use is discussed in Appendix A4.

Ready for more? Take a deep breath and continue on to Chapter 3.

# 微积分代考

## 数学代写|微积分代写Calculus代写|函数的极限

.函数的极限

$f(x)$ 为所有人定义 $x$ 在一段时间内 $x=a$，但不一定在 $x=a$。如果有数字的话 $L$ 对于每一个正数 $\varepsilon$ 对应一个正数 $\delta$ 这样
$$|f(x)-L|<\varepsilon \text { provided } 0<|x-a|<\delta,$$

$$\lim {x \rightarrow a} f(x)=L .$$ 一般的代数运算可以在附录A2所示的极限下进行;因此， $$\lim {x \rightarrow a}[F(x)+G(x)]=\lim {x \rightarrow a} f(x)+\lim {x \rightarrow a} G(x) .$$两个三角极限特别有趣(附录A3):
$$\lim {\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1 \text { and } \lim {\theta \rightarrow 0} \frac{1-\cos \theta}{\theta}=0 .$$下面的极限在微积分中非常有趣，因此给了它一个特殊的名字 $e$，如第109帧和附录A3所述:
$$e=\lim _{x \rightarrow 0}(1+x)^{1 / x}=2.71828 \ldots$$

## 数学代写|微积分代写Calculus代写| differential (frames 264-273)

.微分

$$d y=d\left(x^n\right)=\left(\frac{d}{d x} x^n\right) d x=n x^{n-1} d x .$$

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