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# 数学代写|微积分代写Calculus代考|Theoretical background

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## 数学代写|微积分代写Calculus代考|Theoretical background

Definition 7.1 (Vertical asymptote) Let $\mathcal{A}$ be a subset of $\mathbb{R}$ and $c \in \mathbb{R}$ be a limit point of $\mathcal{A}$. Let $f$ be a function from $\mathcal{A}$ to $\mathbb{R}$. We say that $f$ has a vertical asymptote at the point $c$ if
$$\lim _{x \rightarrow c} f(x)=\infty$$
In this case, we also say that the line of equation $x=c$ is a vertical asymptote.

Definition 7.2 (Horizontal or slant asymptotes at $+\infty$ ) Let $\mathcal{A}$ be a subset of $\mathbb{R}$. Assume that $+\infty$ is a limit point of $\mathcal{A}$ (which means that $\mathcal{A}$ contains arbitrarily large positive numbers). Let $f$ be a function from $\mathcal{A}$ to $\mathbb{R}$. We say that $f$ has an asymptote at $+\infty$ if there exist $m, q \in \mathbb{R}$ such that
$$\lim _{x \rightarrow+\infty}(f(x)-(m x+q))=0 .$$
In this case, we also say that the line of equation $y=m x+q$ is an horizontal asymptote if $m=0$ and a slant asymptote if $m \neq 0$.

Definition 7.3 (Horizontal or slant asymptotes at $-\infty$ ) Let $\mathcal{A}$ be a subset of $\mathbb{R}$. Assume that $-\infty$ is a limit point of $\mathcal{A}$ (which means that $\mathcal{A}$ contains negative numbers, arbitrarily large in modulus). Let $f$ be a function from $\mathcal{A}$ to $\mathbb{R}$. We say that $f$ has an asymptote at $-\infty$ if there exist $m, q \in \mathbb{R}$ such that
$$\lim _{x \rightarrow-\infty}(f(x)-(m x+q))=0 .$$
In this case, we also say that the line of equation $y=m x+q$ is a horizontal asymptote if $m=0$ and a slant asymptote if $m \neq 0$.

It is simple to realize that the line of equation $y=m x+q$ is an asymptote at $+\infty$ if and only if
$$m=\lim {x \rightarrow+\infty} \frac{f(x)}{x} \text {, and } q=\lim {x \rightarrow+\infty}(f(x)-m x) .$$

## 数学代写|微积分代写Calculus代考|Hints on the degree, the asymptotic behaviour and the continuity of an algebraic curve

One of the main aims of this chapter is to show how it is possible, in some simple cases, to draw the graph of an algebraic function (rational or irrational) by using only the notions of degree, asymptote, continuity and the passing of the curve through a few points. By doing so, we shall try to point out the value of these important theoretical notions (degree, continuity) which do not involve much calculus but lead to noteworthy results. For example, we shall consider functions of the type:
$$y=\frac{A(x)}{B(x)}, \quad \text { or } \quad y=\sqrt{C(x)}$$
where $A, B, C$ are polynomials in $x$. These functions can be represented in the following form:
$$\left{\begin{array} { l } { B ( x ) y – A ( x ) = 0 } \ { B ( x ) \neq 0 } \end{array} \text { or } \quad \left{\begin{array}{l} y^2-C(x)=0 \ y \geq 0 \end{array}\right.\right.$$
hence their graphs are given by one or more branches of algebraic curves of the type
$$P(x, y)=0$$
where $P$ is a polynomial in $x, y$ of degree $n$ ( $n$ is called the degree of the curve). For example, in the first case the graph of the function $y=\frac{A(x)}{B(x)}$ is the graph of the curve of equation $B(x) y-A(x)=$ with the exclusion of those point whose abscissas $x$ satisfy the equation $B(x)=0$. In the second case the graph of the function $y=\sqrt{C(x)}$ is the graph of the curve of equation $y^2-C(x)=0$ with the exclusion of those points whose ordinates $y$ satisfy the condition $y<0$.

## 数学代写|微积分代写Calculus代考|Theoretical background

$$\lim _{x \rightarrow c} f(x)=\infty$$

$$\lim _{x \rightarrow+\infty}(f(x)-(m x+q))=0 .$$

$$\lim _{x \rightarrow-\infty}(f(x)-(m x+q))=0 .$$

$$m=\lim {x \rightarrow+\infty} \frac{f(x)}{x} \text {, and } q=\lim {x \rightarrow+\infty}(f(x)-m x) .$$

## 数学代写|微积分代写Calculus代考|Hints on the degree, the asymptotic behaviour and the continuity of an algebraic curve

$$y=\frac{A(x)}{B(x)}, \quad \text { or } \quad y=\sqrt{C(x)}$$

$$\left{\begin{array} { l } { B ( x ) y – A ( x ) = 0 } \ { B ( x ) \neq 0 } \end{array} \text { or } \quad \left{\begin{array}{l} y^2-C(x)=0 \ y \geq 0 \end{array}\right.\right.$$

$$P(x, y)=0$$

## MATLAB代写

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