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# 数学代写|微积分代写Calculus代考|Some Theorems on Limits

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## 数学代写|微积分代写Calculus代考|Some Theorems on Limits

In this appendix we prove several theorems on limits, which show that the usual algebraic manipulations can be carried out with expressions involving limits. We shall show, for example, that
$$\lim {x \rightarrow a}[F(x)+G(x)]=\lim {x \rightarrow a} F(x)+\lim _{x \rightarrow a} G(x)$$
Such results are intuitively reasonable but deserve a formal proof.
Before deriving these theorems, we need to note some general properties of the absolute value function introduced in frame $\mathbf{2 0}$. These properties are
$$\begin{gathered} |a+b| \leq|a|+|b|, \ |a b|=|a||b| . \end{gathered}$$
These relations are easily verified by considering all the possible cases: $a$ and $b$ both negative, both positive, of opposite sign, and one or both equal to zero.

The following are theorems on limits that apply to any two functions $F$ and $G$ such that
$$\lim {x \rightarrow a} F(x)=L \quad \text { and } \lim {x \rightarrow a} G(x)=M$$

## 数学代写|微积分代写Calculus代考|Proof That $\frac{d y}{d x}=\frac{1}{d x / d y}$

If a function is specified by an equation $\gamma=f(x)$, it is ordinarily possible, for at least limited intervals of $x$, to reverse the roles of the dependent and independent variables and to allow the equation to determine the value of $x$ for a given value of $y$. (This cannot always be done, for instance as in the case of the equation $y=a$, where $a$ is a constant.) When such an inversion is possible, the two derivatives are related by
$$\frac{d y}{d x}=\frac{1}{d x / d y} .$$

This relation can be seen as follows:
$$\frac{d y}{d x}=\lim {\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=\lim {\Delta x \rightarrow 0} \frac{1}{\Delta x / \Delta \gamma}=\frac{1}{\lim {\Delta x \rightarrow 0}(\Delta x / \Delta \gamma)}$$ by the limit theorems of Appendix A2. Furthermore, if $\lim {\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x} \neq 0$, then $\Delta y \rightarrow 0$ as $\Delta x \rightarrow 0$, so
$$\frac{d y}{d x}=\frac{1}{\lim _{\Delta y \rightarrow 0}(\Delta x / \Delta y)}=\frac{1}{d x / d y}$$
This result is a further justification of the use of differential notation because normal arithmetic manipulation with differential notation immediately gives
$$\frac{d y}{d x}=\frac{1}{d x / d y}$$

## 数学代写|微积分代写Calculus代考|Some Theorems on Limits

$$\lim x \rightarrow a[F(x)+G(x)]=\lim x \rightarrow a F(x)+\lim _{x \rightarrow a} G(x)$$

$$|a+b| \leq|a|+|b|,|a b|=|a||b| .$$

$$\lim x \rightarrow a F(x)=L \quad \text { and } \lim x \rightarrow a G(x)=M$$

## 数学代写|微积分代写Calculus代考|Proof That $\frac{d y}{d x}=\frac{1}{d x / d y}$

$$\frac{d y}{d x}=\frac{1}{d x / d y}$$

$$\frac{d y}{d x}=\lim \Delta x \rightarrow 0 \frac{\Delta y}{\Delta x}=\lim \Delta x \rightarrow 0 \frac{1}{\Delta x / \Delta \gamma}=\frac{1}{\lim \Delta x \rightarrow 0(\Delta x / \Delta \gamma)}$$

$$\frac{d y}{d x}=\frac{1}{\lim _{\Delta y \rightarrow 0}(\Delta x / \Delta y)}=\frac{1}{d x / d y}$$

$$\frac{d y}{d x}=\frac{1}{d x / d y}$$

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