Limit of functions and operations on limits: senior math course.

The limits of functions are very important to master. The tables below summarize the results you need to know.

These tables are valid for all three situations studied:

  • When the variable x\to,+\infty.
  • When the variable x\to,-\infty.
  • When the variable x\to,a where a \in 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 n\to,+\infty.

The conventions used in these tables are:
l and l' 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

limit sum 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

product limit functions

IV. Limit of the inverse of a function

In the table below, the limit of f equal to 0^+, 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 0^-, but with f(x) < 0.

inverse limit function

V. Limit of a quotient of two functions

We can use: \frac{f}{g}=f\times  ,\frac{1}{g} and with the two previous tables, it is possible to conclude.

In + \infty or in – \infty, 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:

limit quotient function

This table is simplified: ± \infty means + \infty or – \infty.

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.

limits reference function

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