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# 数学代写|微积分代写Calculus代考|One-Sided Limits of Functions

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## 数学代写|微积分代写Calculus代考|One-Sided Limits of Functions

Limits from the Right
We say that the function $f$ has a limit from the right at $\boldsymbol{x}=\boldsymbol{a}$ (or the righthand limit of $f$ exists at $x=a$ ) whose value is $L$ and denote this symbolically by
$$f(a+0)=\lim _{x \rightarrow a^{+}} f(x)=L$$
if BOTH of the following statements are satisfied:

Let $x>a$ and $x$ be very close to $x=a$.

As $x$ approaches $a$ (“from the right” because ” $x>a$ “), the values of $f(x)$ approach the value $L$.
(For a more rigorous definition see the Advanced Topics, later on.)
Table 2.2: One-Sided Limits From the Right
For example, the function $H$ defined by
$$H(x)= \begin{cases}1, & \text { for } x \geq 0 \ 0, & \text { for } x<0\end{cases}$$
called the Heaviside Function (named after Oliver Heaviside, (1850 – 1925) an electrical engineer) has the property that
$$\lim _{x \rightarrow 0^{+}} H(x)=1$$
Why? This is because we can set $a=0$ and $f(x)=H(x)$ in the definition (or in Table 2.2) and apply it as follows:

a) Let $x>0$ and $x$ be very close to 0 ;
b) As $x$ approaches 0 we need to ask the question: “What are the values, $H(x)$, doing?”

Well, we know that $H(x)=1$ for any $x>0$, so, as long as $x \neq 0$, the values $H(x)=1$, (see Fig. 17), so this will be true “in the limit” as $x$ approaches 0 .
Limits from the Left
We say that the function $f$ has a limit from the left at $x=a$ (or the lefthand limit of $f$ exists at $x=a$ ) and is equal to $L$ and denote this symbolically by
$$f(a-0)=\lim _{x \rightarrow a^{-}} f(x)=L$$
if BOTH of the following statements are satisfied:

1. Let $x<a$ and $x$ be very close to $x=a$.
2. As $x$ approaches $a$ (“from the left” because ” $x<a$ “), the values of $f(x)$ approach the value $L$.

## 数学代写|微积分代写Calculus代考|Two-Sided Limits and Continuity

At this point we know how to determine the values of the limit from the left (or right) of a given function $f$ at a point $x=a$. We have also seen that whenever
$$\lim {x \rightarrow a^{+}} f(x) \neq \lim {x \rightarrow a^{-}} f(x)$$
then there is a ‘break’ in the graph of $f$ at $x=a$. The absence of breaks or holes in the graph of a function is what the notion of continuity is all about.
Definition of the limit of a function at $x=a$.
We say that a function $f$ has the (two-sided) limit $L$ as $x$ approaches $a$ if
$$\lim {x \rightarrow a^{+}} f(x)=\lim {x \rightarrow a^{-}} f(x)=L$$
When this happens, we write (for brevity)
$$\lim _{x \rightarrow a} f(x)=L$$
and read this as: the limit of $f(x)$ as $x$ approaches $a$ is $L$ ( $L$ may be infinite here).

NOTE: So, in order for a limit to exist both the right- and left-hand limits must exist and be equal. Using this notion we can now define the ‘continuity of a function $f$ at a point $x=a$.’

We say that $f$ is continuous at $x=a$ if all the following conditions are satisfied:

1. $f$ is defined at $x=a$ (i.e., $f(a)$ is finite)
2. $\lim {x \rightarrow a^{+}} f(x)=\lim {x \rightarrow a^{-}} f(x)(=L$, their common value $)$ and
3. $L=f(a)$.
NOTE: These three conditions must be satisfied in order for a function $f$ to be continuous at a given point $x=a$. If any one or more of these conditions is not satisfied we say that $f$ is discontinuous at $x=a$. In other words, we see from the Definition above (or in Table 2.7) that the one-sided limits from the left and right must be equal in order for $f$ to be continuous at $x=a$ but that this equality, in itself, is not enough to guarantee continuity as there are 2 other conditions that need to be satisfied as well.

## 数学代写|微积分代写Calculus代考|One-Sided Limits of Functions

$$f(a+0)=\lim _{x \rightarrow a^{+}} f(x)=L$$

(有关更严格的定义，请参阅后面的高级主题。)

$$H(x)= \begin{cases}1, & \text { for } x \geq 0 \ 0, & \text { for } x<0\end{cases}$$

$$\lim _{x \rightarrow 0^{+}} H(x)=1$$

a)令$x>0$和$x$非常接近于0;
b)当$x$接近0时，我们需要问这样一个问题:“$H(x)$的值在做什么?”

$$f(a-0)=\lim _{x \rightarrow a^{-}} f(x)=L$$

## 数学代写|微积分代写Calculus代考|Two-Sided Limits and Continuity

$$\lim {x \rightarrow a^{+}} f(x) \neq \lim {x \rightarrow a^{-}} f(x)$$

$$\lim {x \rightarrow a^{+}} f(x)=\lim {x \rightarrow a^{-}} f(x)=L$$

$$\lim _{x \rightarrow a} f(x)=L$$

$f$ 定义于$x=a$(即$f(a)$是有限的)

$\lim {x \rightarrow a^{+}} f(x)=\lim {x \rightarrow a^{-}} f(x)(=L$，它们的共同价值$)$和

$L=f(a)$．

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