# Limits and asymptotes : 12th grade math lesson

Limits (sum, product, quotient) in a 12th grade math lesson with the study of indeterminate forms. In this lesson, we will conduct a study of horizontal, vertical and oblique asymptotes in the 12th grade for the compulsory education.
Knowledge needed for this chapter:
Determine the possible limit of a geometric sequence.
Study the limit of a sum, product or quotient
of two suites.
Use a comparison or framing theorem
to determine a limit of a sequence.
Establish (by derivation or not) the variations of a function.

## I.Limit of a function at infinity

In this section, is the representative curve of the function f in any plane.

### 1. Finite limit in infinity

Definition:
Let f be a function defined at least on an interval of of the type .
The function f has limit ℓ in if any open interval containing ℓ contains all
values of f (x) for x large enough. Then we note: .

Example:

Let f be the function defined on by . We have .
Indeed, the inverse of x approaches 0 as x increases.
Let be an open interval I such that . Then f (x) will always be in I for x large enough.
Graphically, as narrow as a band parallel to the line of equation y = 1 may be, and which
contains, there is always a value of x beyond which does not leave this band.

Horizontal asymptote.
The line with equation y = ℓ is horizontal asymptote to at if .

Remark:

Analogously, we define which characterizes a horizontal asymptote at in of equation y = ℓ.

Example:

We have seen previously that . We also have .
Therefore, the line with equation y = 1 is horizontal asymptote to the curve in and in .

Property (admitted): finite limits of usual functions in ± .
Let n be a non-zero natural number.
and .

## II. Infinite limit in infinity

Definition:
The function f has limit in if any interval of of the type contains
all values of f (x) for x large enough. Then we note: .

Example:

Let f be the square root function. We have.
Indeed, becomes as large as we want as x increases.
Let be an open interval . Then f (x) will always be in I for x large enough.
Graphically, if we consider the upper boundary half-plane a line of equation
y = a, there is always a value of a beyond which does not leave this half-plane.

Property (admitted): infinite limits of usual functions in ±.
Let n be a non-zero natural number.
and

### 2. Infinite limit in a real

Definition:
Let f be a function defined on an open interval of of type or .
The function f has limit in if any interval of of type contains all
the values of f (x) for x close enough to . Then we note: .
Definition: vertical asymptote.
The line of equation is vertically asymptotic to if or .
Property (admitted) : finite limits of usual functions in 0.
Let n be a non-zero natural number.
and

## III. Operations on the limits.

Property: limit of a sum, product and quotient of two functions.

## IV. Limit of a composite function

### 1. Compound function

Definition:
Let f be a function defined on E and having values in F, and let g be a function defined on F.
The compound of f followed by g is the function defined on E by .

Remark:

Do not confuse and which are, in general, different.

### 2. Theorem of composition of limits

Theorem:
Let h be the composite of the function f followed by g and a, b and c three real or ± .
If and , then .

## V. Limitations and comparison

Theorem:

### 2. The so-called “gendarmes” or “sandwich” framing theorem.

Theorem:
Let there be two real a and ℓ and three functions f , g and h such that, for x > a, we have .
If , then .

Remark:

As for the previous comparison theorem, we have two theorems
analogous when x tends to – and when x tends to a real .

Example:

Let’s determine the limit in – of .
The limit of cos x in – is indeterminate. Therefore the one of f (x) too.
However, for any strictly negative real x, so .
And dividing member by member by we have :
.

For ,.

Now, .

So, according to the gendarme theorem,.

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