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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.
I. Limit of a sum of two functions
II. Limit of a difference of two functions
Use: f – g = f + (-g) and the previous table.
III. Limit of a product of two functions
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.
VI. Limit of reference functions.
VII. The comparison theorems
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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