A eleventh math lesson on the limits of functions and the existence of an asymptote to the representative curve of this function. The limits of functions and the study of horizontal, vertical and oblique asymptotes in a 11th grade math lesson where we will discuss the definition of the asymptote of a curve. In this first grade lesson, we will see the different operations on limits and the comparison theorem.

The tables below summarize the results you need to know.

These tables are valid for all three situations studied:

• When the variable .
• When the variable .
• When the variable where a R.

But it goes without saying that, for the two functions f and g concerned, the limits are taken at the same place!
In the particular case where the functions are numerical sequences, we can use these results by replacing f by (Un) and g by (Vn) with the only possible case the variable .

The conventions used in these tables are:
and are real numbers (finite limits).
– ? indicates that in this situation there is no general conclusion.

It is sometimes said to be an ” indeterminate form ” noted F.I.

In these cases, it will be necessary to develop other methods of resolution.

## II. Limit of a difference of two functions

Use: f – g = f + (-g) and the previous table.

## IV. Limit of the inverse of a function

In the table below, the limit of f equal to , means that, at the point where the limit is taken, this limit is zero and that, for any x close enough to this point, we have f(x) > 0.
Similar definition for , but with f(x) < 0.

## V. Limit of a quotient of two functions

We can use: and with the two previous tables, it is possible to conclude.

In + or in – , the limit of a rational function is the limit of the quotient of the highest degree terms of the numerator and the denominator.

The following results can also be noted:

This table is simplified: ± means + or – .

To decide, we apply the rule of the sign of the quotient according to the signs of f and g in the neighborhood of the place where the limit is sought.

## VII. The comparison theorems

Theorem:

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

When a = – The functions are defined on R or an interval I of the form ] – ; A ] where A is a real number.
When a R , functions are defined on R or an interval I of the form [ A ; B ] where A and B are real and a[ 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 ] – ; 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 ; + [ 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 (which will often be 0).

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