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**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

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.

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 .

and .

## II. Infinite limit in infinity

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.

and

### 2. Infinite limit in a real

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: .

and

## III. Operations on the limits.

## IV. Limit of a composite function

### 1. Compound function

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

If and , then .

## V. Limitations and comparison

### 1. Comparison theorem

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

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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