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# 数学代写|实分析代写Real analysis代考|MATH351 L’Hospital’s Rule

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## 数学代写|实分析代写Real analysis代考|L’Hospital’s Rule

L’Hospital’s rule is useful for evaluating limits of the form
$$\lim {x \rightarrow p} \frac{f(x)}{g(x)}$$ where either (a) $\lim {x \rightarrow p} f(x)=\lim {x \rightarrow p} g(x)=0$ or (b) $f$ and $g$ tend to $\pm \infty$ as $x \rightarrow p$. If (a) holds, then $\lim {x \rightarrow p}(f(x) / g(x))$ is usually referred to as indeterminate of form $0 / 0$, whereas in (b) the limit is referred to as indeterminate of form $\infty / \infty$. The reason that (a) and (b) are indeterminate are that previous methods may no longer apply.

In (a), if either $\lim {x \rightarrow p} f(x)$ or $\lim {x \rightarrow p} g(x)$ is nonzero, then previous methods discussed in Section $4.1$ apply. For example, if both $f$ and $g$ have limits at $p$ and $\lim {x \rightarrow p} g(x) \neq 0$, then by Theorem 4.1.6(c) $$\lim {x \rightarrow p} \frac{f(x)}{g(x)}=\frac{\lim {x \rightarrow p} f(x)}{\lim {x \rightarrow p} g(x)} .$$
On the other hand, if $\lim {x \rightarrow p} f(x)=A \neq 0$ and $g(x)>0$ with $\lim {x \rightarrow p} g(x)=0$, then as $x \rightarrow p, f(x) / g(x)$ tends to $\infty$ if $A>0$, and to $-\infty$ if $A<0$ (Exercise 5). However, if $\lim {x \rightarrow p} f(x)=\lim {x \rightarrow p} g(x)=0$, then unless the quotient $f(x) / g(x)$ can somehow be simplified, previous methods may no longer be applicable.

## 数学代写|实分析代写Real analysis代考|Newton’s Method

In this section, we consider the iterative method, commonly known as Newton’s method, for finding approximations to the solutions of the equation $f(x)=0$. Although the method is named after Newton, it is actually due to Joseph Raphson (1648-1715) and in many texts the method is referred to as the Newton-Raphson method. Newton did derive an iterative method for finding the roots of a cubic equation; his method however is not the one used in the procedure named after him. That was developed by Raphson.

Suppose $f$ is a continuous function on $[a, b]$ satisfying $f(a) f(b)<0$. Then $f$ has opposite sign at the endpoints $a$ and $b$ and thus by the intermediate value theorem (Theorem 4.2.11) there exists at least one value $c \in(a, b)$ for which $f(c)=0$. If in addition $f$ is differentiable on $(a, b)$ with $f^{\prime}(x) \neq 0$ for all $x \in(a, b)$, then $f$ is either strictly increasing or decreasing on $[a, b]$, and in this case the value $c$ is unique; that is, there is exactly one point where the graph of $f$ crosses the $x$-axis.

An elementary approach to finding a numerical approximation to the value $c$ is the method of bisection. For this method, differentiability of $f$ is not required. To illustrate the method, suppose $f$ satisfies $f(a)<00$. Then $c \in\left(a, c_{1}\right)$, and in this case we set $c_{0}=a$ and
$$c_{2}=\frac{1}{2}\left(c_{0}+c_{1}\right) .$$
If $f\left(c_{2}\right)=0$, we are done. If not, then suppose $f\left(c_{2}\right)<0$. Then $c \in\left(c_{2}, c_{1}\right)$, and as above we set $$c_{3}=\frac{1}{2}\left(c_{1}+c_{2}\right) .$$ In general, suppose $c_{1}, c_{2}, \ldots, c_{n}, n \geq 2$, have been determined. If by happenstance $f\left(c_{n}\right)=0$, then we have obtained the exact value. If $f\left(c_{n-1}\right) f\left(c_{n}\right)<0$, then $c$ lies between $c_{n-1}$ and $c_{n}$, and we define $$c_{n+1}=\frac{1}{2}\left(c_{n}+c_{n-1}\right) .$$ On the other hand, if $f\left(c_{n-1}\right) f\left(c_{n}\right)>0$, then $c$ lies between $c_{n}$ and $c_{n-2}$, and in this case, we define
$$c_{n+1}=\frac{1}{2}\left(c_{n}+c_{n-2}\right)$$

## 数学代写|实分析代写Real analysis代考|L’Hospital’s Rule L’Hospital

$$\lim x \rightarrow p \frac{f(x)}{g(x)}$$

$\lim x \rightarrow p(f(x) / g(x))$ 通常被称为形式不定 $0 / 0$ ，而在 (b) 中，极限被称为形式不定 $\infty / \infty$. (a) 和 (b) 不确定的原因是以前 的方法可能不再适用。

$$\lim x \rightarrow p \frac{f(x)}{g(x)}=\frac{\lim x \rightarrow p f(x)}{\lim x \rightarrow p g(x)} .$$

## 数学代写|实分析代写Real analysis代考|Newton’s Method

$$c_{n+1}=\frac{1}{2}\left(c_{n}+c_{n-2}\right)$$

## MATLAB代写

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