微积分作业代写calclulus代考| Limits and continuity

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微积分作业代写calclulus代考|Limit laws

If $\lim {x \rightarrow a} f(\boldsymbol{x})=L, \quad \lim {\boldsymbol{x} \rightarrow a} g(\boldsymbol{x})=M$, then the following sum, product, quotient, convergence and composition results can be proved.
(a) $\lim {\boldsymbol{x} \rightarrow a}(f(\boldsymbol{x})+g(\boldsymbol{x}))=L+M$ (b) $\lim {\boldsymbol{x} \rightarrow a}(f(\boldsymbol{x}) . g(\boldsymbol{x}))=L \cdot M$
(c) $\lim {\boldsymbol{x} \rightarrow a} \frac{f(\boldsymbol{x})}{g(\boldsymbol{x})}=\frac{L}{M} \quad(M \neq 0)$ (d) $\lim {\boldsymbol{x} \rightarrow a} f(\boldsymbol{x})=\lim {\boldsymbol{x} \rightarrow a} g(\boldsymbol{x})$ and $f(\boldsymbol{x}) \leq h(\boldsymbol{x}) \leq g(\boldsymbol{x})$ means that $\lim {\boldsymbol{x} \rightarrow a} h(\boldsymbol{x})$ exists and equals $L$ which equals $M$ (a “squeeze theorem”).
(e) If $F(t)$ is a continuous function at $t=L$ then
$$\lim {x \rightarrow a} F(f(x))=F(L)=F\left(\lim {x \rightarrow a} f(x)\right)$$
That is, for continuous functions, we may interchange the limit and function composition operations.

微积分作业代写calclulus代考|Example 2.4:

Consider $L=\lim {(x, y) \rightarrow(1, \pi)} \frac{\cos (x y)}{1-x-\cos y}=\lim {(x, y)} \frac{g(x, y)}{h(x, y)}$, noting in particular that $\lim {(x, y) \rightarrow(1, \pi)} h(x, y) \neq 0$. Applying the standard rules we find that $$L=\frac{\lim \cos (x y)}{\lim (1-x-\cos y)}=\frac{-1}{+1}=-1$$ Here, we have used the sum, product, quotient, and composition laws. In evaluating limits of any well-behaved $f: \mathbb{R}^{n} \longrightarrow \mathbb{R}$ for $n>2$, we follow the exact same process as implied in the above example: besides using the limit laws, the reader can also make use of results from the study of limits of functions of one variable, some of which are listed on Page 24. However, the simple statement made in the limit definition hides considerable detail that we need to confront in more complicated cases. Definition $2.2$ implicitly means that $* \lim {\boldsymbol{x} \rightarrow a} f(\boldsymbol{x})$ exists and is equal to $L$ if $f \longrightarrow L$ independently of how $\boldsymbol{x}$ approaches $\boldsymbol{a}$ !

• The limit $L$, if it exists, is unique!
• No limit of $f$ exists if $f$ has different limits when $\boldsymbol{x}$ approaches $\boldsymbol{a}$ along different curves!

The graphical depiction of Definition 2.2, in analogy with Figure $1.16$, is shown in Figure $2.4$ on the next page.

微积分作业代写calclulus代考|Limit laws

\lim {x \rightarrow a} F(f(x))=F(L)=F\left(\lim {x \rightarrow a} f(x)\right)


微积分作业代写calclulus代考|Example 2.4:

• 限制一世，如果存在，是唯一的！
• 没有限制F如果存在F有不同的限制时X方法一种沿着不同的曲线！