The Limit of Sequences and Functions: A Grade 12 Math Course PDF

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1 I. Suites and functions: comparative study.
- 1.1 1 Sense of variation of functions and numerical sequences.
- 1.2 2. major, minor and bounded sequences and functions.
2 II. limits of sequences and functions.
3 II.case where the function or the sequence has no limit.
4 III. The comparison theorems

The limit of sequences and functions is very important in mathematics. In addition, it requires regular practice of various exercises on the chapter.

In addition, with a free downloadable senior math course in PDF format, you will be more comfortable.

In addition, a comparative study of numerical sequences and functions is essential for senior students. In addition, the study of the limit of a sequence and a function at infinity or at a finite value is also to be understood to make a good progression in math.

Remember to take this course to take advantage of its benefits.

I. Suites and functions: comparative study.

Remark:

The numerical sequences being particular functions (defined on N or a part of N ), one will find necessarily for the sequences and the functions, analogous properties.

Not all properties will be listed here, but only those whose comparison is instructive.

The sequences will appear in the right column and the numerical functions in the left column. Except in special cases, sequences are defined on N or possibly a part of N (from a certain rank ) and functions are defined on R or a part of R (most of the time on an interval I ).

1 Sense of variation of functions and numerical sequences.

f increasing on I $\Leftrightarrow$ For all a $\in$ I and b $\in$ I, if a $\leq\,$ b then f(a) $\leq\,$ f(b)

f strictly increasing on I $\Leftrightarrow$ For all a $\in$ I and b $\in$ I, if a < b then f(a) < f(b)

f decreasing on I $\Leftrightarrow$ For all a $\in$ I and b $\in$ I, if a $\leq\,$ b then f(a) $\geq\,$ f(b)

f strictly decreasing on I $\Leftrightarrow$ For all a $\in$ I and b $\in$ I, if a < b then f(a) > f(b)

f constant on I $\Leftrightarrow$ For all a $\in$ I and b $\in$ I, if a $\leq\,$ b then f(a) = f(b)

f monotonic on I $\Leftrightarrow$ f keeps the same direction of variation on the interval I

If f is derivable on I, then: f increasing on I For any x $\in$ I, f ‘ (x) $\geq\,$ 0

If f is differentiable on I, then: f decreasing on I For any x $\in$ I, f ‘ (x) $\leq\,$ 0

If f is derivable on I, then: f constant on I For any x $\in$ I, f ‘ (x) = 0

(_Un) increasing For all n $\in$ N , we have:_Un $\leq\,$ _Un+1

(_Un) strictly increasing For all n $\in$ N , we have:_Un < _Un+1

(_Un) decreasing For all n $\in$ N , we have:_Un $\geq\,$ _Un+1

(_Un) strictly decreasing For all n $\in$ N , we have:_Un > _Un+1

(_Un) constant For all n $\in$ N , we have:_Un =_Un+1

(_Un) monotonic (_Un) is either increasing, decreasing or constant.

Note:

It may be that this is not true for the whole set N : specify then from which rank this is true.

2. major, minor and bounded sequences and functions.

f major on I $\Leftrightarrow$ There exists M $\in$ R, such that, for any x $\in$ I, we have: f(x) $\leq\,$ M

f minor on I $\Leftrightarrow$ There exists m $\in$ R, such that, for any x $\in$ I, we have: f(x) $\geq\,$ m

f bounded on I $\Leftrightarrow$ f is increased and decreased on I

(_Un) increased $\Leftrightarrow$ There exists M $\in$ R, such that, for any n $\in$ N, we have:_Un $\leq\,$ M

(_Un) minor $\Leftrightarrow$ There exists m $\in$ R, such that, for any x $\in$ N, we have:_Un $\geq\,$ m

(_Un) bounded $\Leftrightarrow$ (_Un) is increased and decreased

II. limits of sequences and functions.

1.finite limit (l $\in$ $\mathbb{R}$ ) and theimite in + $\infty$ of functions and sequences.

2.finite limit in – $\infty$ of numerical functions.

Ownership:

If f is defined on R or an interval of the form ]- $\infty$ ; A] where A $\in$ R and if any open interval containing also contains all the values of f(x) for – x large enough, we say that f tends to when x tends to – $\infty$ .

We write $\lim_{x\to,-\infty}f(x)=l$

In this situation, the graphical representation of f has a horizontal asymptote of equation y = in – $\infty$ .

Example:

$\lim_{x\to,-\infty},\frac{sinx}{x}=0$ .

3. finite limit of a function at a point a $\in$ R

Ownership:

If f is defined on R or an interval I containing a $\in$ R and if any open interval containing also contains all values of f(x) for any x $\in$ I close enough to a, we say that f tends to when x tends to a.

We write $\lim_{x\to,a}f(x)=l$ .

Example:

$\lim_{x\to,0},\frac{sinx}{x}=1$

II.case where the function or the sequence has no limit.

Without going into theoretical details, we will give some examples of functions and sequences that do not have a limit. It is interesting to visualize these examples on a graphing calculator.

1. f defined on $\mathbb{R}^*$ by $f(x)=\frac{,%7Cx,,%7C}{x}$ .

We have: If x > 0, f(x) = 1, so $\lim_{x\to,0^+}f(x)=1$ and if x < 0, f(x) = -1, so $\lim_{x\to,0^+}f(x)=-1$ .

The limits to the left and to the right of 0 exist, but are different so f has no limit in 0.

2. For any n $\in$ N,_Un = sin n takes an infinity of values on the interval [-1 ; 1], without ever approaching a limit value.

3. For any n $\in$ N, takes alternatively the values -1 and 1, thus no limit.

III. The comparison theorems

Theorem:

For functions, in the properties below, the letter a designates a real number as well as +. $\infty$ or – $\infty$ .
When a = + $\infty$ The functions are defined on R or an interval I of the form [ A ; + ]. $\infty$ [ where A is a real number.

When a = – $\infty$ The functions are defined on R or an interval I of the form ] – $\infty$ ; A ] where A is a real number.
When a $\in$ R , functions are defined on R or an interval I of the form [ A ; B ] where A and B are real and a $\in$ [ A ; B ].
If the limit in question is the left-hand limit of a, the functions are defined on an interval I of the form ] – $\infty$ ; a [ or [ A ; a [ where A is a real number.
If the limit concerned is the limit to the right of a, the functions are defined on an interval I of the form ] a ; + $\infty$ [ or ] a ; A ] where A is a real number.

For sequences, the index n is a natural number greater than or equal to a certain rank n_0 (which will often be 0).

IV. Composition theorem of two functions:

Theorem:

If f is a function defined on an interval J such that, for any x $\in$ I, we have:
y = u(x) $\in$ J, i.e.: u(I) $\subset$ J.
If we also have $\lim_{x\to,a}u(x)=b$ and $\lim_{y\to,b}f(y)=l$ , then $\lim_{x\to,a}fou(x)=,l$ .

If f is a function defined on an interval J such that:
for any integer $n\geq\,,n_0$ we have: $u_n\in,J$
If we also have $\lim_{n\to,+\infty}u_n=b$ and $\lim_{y\to,b}f(y)=l$ , then $\lim_{n\to,+\infty}f(u_n)=l$ .

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If f is defined on R or an interval of the form [A; +], then $\infty$ [ where A $\in$ R and if any open interval containing

also contains all values of f(x) for x large enough, we say that f tends to

when x tends to +. $\infty$ .we write $\lim_{x\to,+\infty}f(x)=l$ .

In this situation, the graphical representation of f has a horizontal asymptote of equation y = in +. $\infty$ .

Example:

$\lim_{x\to,+\infty},\frac{2x+1}{x-1}=2$

If any open interval containing

also contains all the terms of the sequence (Un) from a certain rank, we say that (_Un) converges to

. We write $\lim_{n\to,+\infty}U_n=l$

Example:

$\lim_{n\to,+\infty}5-\frac{1}{2^n}=5$

I. Suites and functions: comparative study.

1 Sense of variation of functions and numerical sequences.

2. major, minor and bounded sequences and functions.

II. limits of sequences and functions.

2.finite limit in – <img decoding="async" class="LatexImg" src="https://mathematiques-web.fr/cgi-bin/mimetex.cgi?\infty" alt="\infty" alt="\infty" > of numerical functions.

3. finite limit of a function at a point a<img decoding="async" class="LatexImg" src="https://mathematiques-web.fr/cgi-bin/mimetex.cgi?\in" alt="\in" alt="\ín" >R

II.case where the function or the sequence has no limit.

III. The comparison theorems

D'autres fiches dans la section High school math course

D'autres articles analogues à limit of sequences and functions: high school math course in PDF

2.finite limit in – $\infty$ of numerical functions.

3. finite limit of a function at a point a $\in$ R