# 微积分网课代修|微分学代写Differential calculus代考|MTH295 The definition of limit

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## 微积分网课代修|微分学代写Differential calculus代考|The definition of limit

Calculus, in large part, is the study of how to properly handle infinity.
Example 1.3.1: something from nothing?
Let’s examine this seemingly legitimate computation:
That’s impossible! What, $0=1$ ? How did this happen?
Consider a little story below:

The numbers refer to the amount of soil taken out, and one can say that we got something from nothing!
Our mistake was to be too casual about carrying out infinitely many algebraic operations.

Which of the ” $=”$ signs above is incorrect? Hint: Think of this computation as a process.
The question has been:
-With this sequence, what number do its values approach?
We can also turn this around:
With this number, the values of what sequence approach it?

So, the limit is a number and the sequence approximates this number:
$\begin{array}{lllllllll}\text { (1) } & \text { The sequence } & 1 & 1 / 2 & 1 / 3 & 1 / 4 & 1 / 5 & \ldots & \text { approximates } 0 . \ \text { (2) } \text { The sequence } & .9 & .99 & .999 & .9999 & 99999 & \ldots & \text { approximates } 1 . \ \text { (3) The sequence } & 1 . & 1.1 & 1.01 & 1.001 & 1.0001 & \ldots & \text { approximates } 1 . \ \text { (4) } & \text { The sequence } & 3 . & 3.1 & 3.14 & 3.141 & 3.1415 & \ldots & \text { approximates } \pi . \ \text { (5) } & \text { The sequence } & 1 & 2 & 3 & 4 & 5 & \ldots & \text { approaches } \infty . \ \text { (6) } & \text { The sequence } & 0 & 1 & 0 & 1 & 0 & \ldots & \text { doesn’t approximate any number. }\end{array}$
So, we can substit ute the sequence for the number it approximates and do it with any degree of accuracy!
We use the following notation for the limit of a sequence:
Limit of sequence
$$a_{n} \rightarrow a$$
We can rewrite the above:
\begin{tabular}{llllllll|l}
list & \multicolumn{10}{c|}{} & & & & \
\hline$(1)$ & 1 & $1 / 2$ & $1 / 3$ & $1 / 4$ & $1 / 5$ & $\ldots$ & $\rightarrow 0$ & $1 / n \rightarrow 0$ \
$(2)$ & $.9$ & $.99$ & $.999$ & 9999 & 99999 & $\ldots$ & $\rightarrow 1$ & $1-10^{-n} \rightarrow 1$ \
$(3)$ & $1 .$ & $1.1$ & $1.01$ & $1.001$ & $1.0001$ & $\ldots$ & $\rightarrow 1$ & $1+10^{-n} \rightarrow 1$ \
$(4)$ & $3 .$ & $3.1$ & $3.14$ & $3.141$ & $3.1415$ & $\ldots$ & $\rightarrow \pi$ & \
$(5)$ & 1 & 2 & 3 & 4 & 5 & $\ldots$ & $\rightarrow+\infty$ & $n \rightarrow+\infty$ \
$(6)$ & 0 & 1 & 0 & 1 & 0 & $\ldots$ & $\rightarrow$ nothing &
\end{tabular}
Now, let’s find the exact meaning of limit.

## 微积分网课代修|微分学代写Differential calculus代考|Limits under algebraic operations

If every real number is the sequence of its approximations, does the usual arit hmet ic operations with numbers match those with the corresponding sequences? Yes. We will discover the following:
Limits behave well with respect to algebra.
For simplicity, we assume below that all the sequences are defined on the same set of integers.
Ex ample 1.4.1: algebra of sequences
What do we mean by adding, multiplying, etc. two sequences? Just as with functions, we add, multiply, etc. term-wise:
\begin{tabular}{lllllll}
$n$ & 1 & 2 & 3 & $\ldots$ & $n$ & $\ldots$ \
\hline$a_{n}$ & 1 & $1 / 2$ & $1 / 3$ & $\ldots$ & $1 / n$ & $\ldots$ \
$b_{n}$ & $-1$ & 1 & $-1$ & $\ldots$ & $(-1)^{n}$ & $\ldots$ \
\hline$a_{n}+b_{n}$ & $1-1$ & $1 / 2+1$ & $1 / 3-1$ & $\ldots$ & $1 / n+(-1)^{n}$ & $\ldots$ \
$a_{n} \cdot b_{n}$ & $1 \cdot 1$ & $1 / 2 \cdot 1$ & $1 / 3 \cdot(-1)$ & $\ldots$ & $1 / n \cdot(-1)^{n}$ & $\ldots$
\end{tabular}
Of course, we really need only the $n$th column.

## 微积分网课代修|微分学代写Differential calculus代考|The definition of limit

(1) The sequence $1 \quad 1 / 2 \quad 1 / 3 \quad 1 / 4 \quad 1 / 5 \quad \ldots \quad$ approximates 0 . (2) The sequen 所以，我们可以用序列代替它近似的数字，并以任何程度的准确度来做!

$$a_{n} \rightarrow a$$