### 数学代写|微积分代写Calculus代写|MATH7000

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|微积分代写Calculus代写|Left Limit and Right Limit

It can happen that $\lim {x \rightarrow a} f(x)$ does not exist, but $f(x)$ can approach a particular value as $x$ approaches $a$ either from left or from right, as in the following figure corresponding to Example 2.1.3 (Fig. 2.8). Definition 2.1.5 Let $f$ be a real valued function defined on a set $D \subseteq \mathbb{R}$. (1) Suppose $D \cap(-\infty, a) \neq \varnothing$ and $a$ is a limit point of $D \cap(-\infty, a)$. Then we say that $f(x)$ has the left limit $b \in \mathbb{R}$ as $x$ approaches $a \in \mathbb{R}$ from left if for every $\varepsilon>0$, there exists $\delta>0$ such that $$|f(x)-b|<\varepsilon \quad \forall x \in D \cap(a-\delta, a),$$ and in that case we write $\lim {x \rightarrow a^{-}} f(x)=b$.
(2) Suppose $D \cap(a, \infty) \neq \varnothing$ and $a$ is a limit point of $D \cap(a, \infty)$. Then we say that $f(x)$ has the right limit $b \in \mathbb{R}$ as $x$ approaches $a \in \mathbb{R}$ from right if for every $\varepsilon>0$, there exists $\delta>0$ such that
$$|f(x)-b|<\varepsilon \quad \forall x \in D \cap(a, a+\delta)$$
and in that case we write $\lim {x \rightarrow a^{+}} f(x)=b$. Notation 2.1.3 We shall use the notations: $$f(a-):=\lim {x \rightarrow a^{-}} f(x), \quad f(a+):=\lim _{x \rightarrow a^{+}} f(x)$$
whenever the above limits exist.

## 数学代写|微积分代写Calculus代写|Limit at ±∞ and Limit ±∞

Definition 2.1.6 Let $f$ be a real valued function defined on $D_{f} \subseteq \mathbb{R}$.
(1) If $D_{f}$ contains $(a, \infty)$ for some $a \in \mathbb{R}$, then $f$ is said to have the limit $b$ as $x \rightarrow \infty$, if for every $\varepsilon>0$, there exists $M>a$ such that
$$|f(x)-b|<\varepsilon \text { whenever } x>M,$$
and in that case we write $\lim {x \rightarrow \infty} f(x)=b$. (2) If $D{f}$ contains $(-\infty, a)$ for some $a \in \mathbb{R}$, then $f$ is said to have the limit $b$ as $x \rightarrow-\infty$, if for every $\varepsilon>0$, there exits $M<a$ such that
$$|f(x)-b|<\varepsilon \quad \text { whenever } x<M,$$
and in that case we write $\lim {x \rightarrow-\infty} f(x)=b .$ Notation 2.1.4 When we write $\lim {x \rightarrow \infty} f(x)=b$ we mean that $D_{f}$ contains an interval of the form $(a, \infty)$ for some $a \in \mathbb{R}$ and the limit is in the sense of Definition 2.1.6(1). Similarly, when we write we $\lim {x \rightarrow-\infty} f(x)=b$, it is assumed that $D{f}$ contains an interval of the form $(-\infty, a)$ for some $a \in \mathbb{R}$, and the limit is in the sense of Definition 2.1.6(2).

Also, for a sequence $\left(x_{n}\right)$ in $\mathbb{R}$, if we write $f\left(x_{n}\right) \rightarrow b$ for some $b \in \mathbb{R}$, we assume that $x_{n}$ belongs to $D_{f}$, and the limit of $\left(f\left(x_{n}\right)\right)$ is $b$.

Now, we give the sequential characterizations of limits as given in Definition $2.1 .6$.
Theorem 2.1.13 The following hold.
(i) $\lim {x \rightarrow \infty} f(x)=b$ if and only if for every sequence $\left(x{n}\right), x_{n} \rightarrow \infty$ implies $f\left(x_{n}\right) \rightarrow b$

(ii) $\lim {x \rightarrow-\infty} f(x)=b$ if and only if for every sequence $\left(x{n}\right), x_{n} \rightarrow-\infty$ implies $f\left(x_{n}\right) \rightarrow b$

## 数学代写|微积分代写Calculus代写|Definition and Some Basic Results

Definition 2.2.1 Let $f$ be a real valued function defined on a set $D \subseteq \mathbb{R}$. Then $f$ is said to be continuous at a point $x_{0} \in D$ if for every $\varepsilon>0$, there exists a $\delta>0$ such that
$$\left|f(x)-f\left(x_{0}\right)\right|<\varepsilon \text { whenever } x \in D,\left|x-x_{0}\right|<\delta .$$
The function $f$ is said to be continuous on $D$ if it is continuous at every point in D.

Note that, in the above definition, we did not assume that $x_{0}$ is a limit point of D. However, if we assume that $x_{0}$ is a limit point of $D$, then we have the following characterization of continuity.

Theorem 2.2.1 Let $x_{0} \in D$ be a limit point of $D$. Then, for a function $f: D \rightarrow \mathbb{R}$, the following are equivalent.
(i) $f$ is continuous at $x_{0}$.
(ii) $\lim {x \rightarrow x{0}} f(x)$ exists and it is equal to $f\left(x_{0}\right)$.
(iii) For every sequence $\left(x_{n}\right)$ in $D, x_{n} \rightarrow x_{0}$ implies $f\left(x_{n}\right) \rightarrow f\left(x_{0}\right)$.
Recall that a point $x_{0}$ in an interval $I$ is a limit point of $I$ if and only if either $x_{0} \in I$ or if $x_{0}$ is an endpoint of $I$. In this book, we shall consider continuity of functions which are defined on intervals.

Convention: When we say that $f$ is continuous at a point $x_{0} \in \mathbb{R}$, we mean that $f$ is defined on an interval containing $x_{0}$ and $f$ is continuous at $x_{0}$.

Example 2.2.1 Let $f(x)=a_{0}+a_{1} x+\cdots a_{k} x^{k}$ for some $a_{0}, a_{1}, \ldots, a_{k}$ in $\mathbb{R}$ and $k \in \mathbb{N}$. We know (cf. Example 2.1.8) that, for any $x_{0} \in \mathbb{R}$,
$$\lim {x \rightarrow x{0}} f(x)=f\left(x_{0}\right)$$
Hence, by Theorem 2.2.1, $f$ is continuous at every $x_{0} \in \mathbb{R}$ (Fig. 2.9).
Example 2.2.2 Let $f:[-1,1] \rightarrow \mathbb{R}$ be as in Example 2.1.3, i.e.,
$$f(x)=\left{\begin{array}{l} 0,-1 \leq x \leq 0 \ 1,0<x \leq 1 \end{array}\right.$$
We have seen that $\lim {x \rightarrow 0} f(x)$ does not exist. Hence, by Theorem 2.2.1, $f$ is not continuous at 0 . Example 2.2.3 We have seen in Examples 2.1.10 and 2.1.11 that $$\lim {x \rightarrow 0} \sin x=0, \quad \lim {x \rightarrow 0} \cos x=1, \quad \lim {x \rightarrow 0} \frac{\sin x}{x}=1 .$$

## 数学代写|微积分代写Calculus代写|Left Limit and Right Limit

|F(X)−b|<e∀X∈D∩(一个−d,一个),在这种情况下，我们写林X→一个−F(X)=b.
(2) 假设D∩(一个,∞)≠∅和一个是一个极限点D∩(一个,∞). 然后我们说F(X)有正确的限制b∈R作为X方法一个∈R从右边开始，如果对于每个e>0， 那里存在d>0这样

|F(X)−b|<e∀X∈D∩(一个,一个+d)

F(一个−):=林X→一个−F(X),F(一个+):=林X→一个+F(X)

## 数学代写|微积分代写Calculus代写|Limit at ±∞ and Limit ±∞

(1) 如果DF包含(一个,∞)对于一些一个∈R， 然后F据说有极限b作为X→∞, 如果对于每个e>0， 那里存在米>一个这样

|F(X)−b|<e 每当 X>米,

|F(X)−b|<e 每当 X<米,

（一世）林X→∞F(X)=b当且仅当对于每个序列(Xn),Xn→∞暗示F(Xn)→b

(二)林X→−∞F(X)=b当且仅当对于每个序列(Xn),Xn→−∞暗示F(Xn)→b

## 数学代写|微积分代写Calculus代写|Definition and Some Basic Results

|F(X)−F(X0)|<e 每当 X∈D,|X−X0|<d.

（一世）F是连续的X0.
(二)林X→X0F(X)存在并且它等于F(X0).
(iii) 对于每个序列(Xn)在D,Xn→X0暗示F(Xn)→F(X0).

$$f(x)=\left{ 0,−1≤X≤0 1,0<X≤1\正确的。 在和H一个在和s和和n吨H一个吨林X→0F(X)d○和sn○吨和X一世s吨.H和nC和,b是吨H和○r和米2.2.1,F一世sn○吨C○n吨一世n在○在s一个吨0.和X一个米pl和2.2.3在和H一个在和s和和n一世n和X一个米pl和s2.1.10一个nd2.1.11吨H一个吨\lim {x \rightarrow 0} \sin x=0, \quad \lim {x \rightarrow 0} \cos x=1, \quad \lim {x \rightarrow 0} \frac{\sin x}{x} =1。$$

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## MATLAB代写

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