!! used as default html header if there is none in the selected theme. OEF Limit calculus with logarithms or exponentials

# OEF Limit calculus with logarithms or exponentials --- Introduction ---

This module contains 7 exercises about the limit calculus of logarithm and exponential functions. The required and tested skills are:
• limits of polynoms and quotient of polynoms, of functions ln and exp ;
• computational properties of limits (theorems about the limits of sums, products, quotients, composed functions) ;
• indeterminate forms;
• compared growth properties between polynoms and the functions exp and ln.
The exercises are composed of several steps. An exercise goes on, even if a false reply has been given at the precedent step. The good answers are provided after each step, to enable further evaluations. NEW EXERCISES. PLEASE SIGNAL ANY BUG...

### Limit of u(x)*exp(kx)

We consider the function defined over .
The aim of the exercise is to compute step by step the limits of , at and at respectively.

• Let be the function defined over .
Evaluate the limits of at and at : ( )
=   and   =
• The limits of at and at are:
and
• Now evaluate the limits of at and at : ( )
=   and   =
• The limits of the exponential function at and at are:
and
• From the preceding results, one can deduce the limit of at by using =
• From the preceding results, and , one deduces that:
• From the preceding results, one can deduce the limit of at by using =

### Limit of u(x)*ln(kx)

Let us consider the function defined over .
The aim of the exercise is to evaluate step by step the limit of , at and at respectively.

• Let be the function defined over .
Evaluate the limits of at and at : ( )
=   and   =
• The limits of at and at are:
and
• Evaluate now the limits of at and at : ( )
=   and   =
• The limits of the logarithm at and at are:
and
• From the preceding results, one can deduce the limit of at by applying the =
• From the preceding results, by the , it comes that:
• From the preceding results, one can deduce the limit of at by applying the =

### Limit of k.ln(ax+b) or k/ln(ax+b)

Let be the function defined over by: .

The aim of the exercise is to evaluate step by step the limit of at .

• The function is of the form with:
= and =
• The function is of the form with and .
• Evaluate the limit of at : ( )
=
• The limit of at is:
• Evaluate the limit of ar )
=
• From the properties of the logarithm function, we know that:
• By variable renaming and by composition of limits, it comes that: ( )
=
• By composition, the limit of at is:
.
• Eventually, by the computational rules of the limits, it comes that: ( )
=

### Limit of k.exp(ax+b) or k/exp(ax+b)

Let be the function defined over by: .

The aim of the exercise is to evaluate step by step the limit of at .

• The function is of the form with:
= and =
• The function is of the form with and .
• Evaluate the limit of at : ( )
=
• The limit of at is:
• Evaluate the limit of at )
=
• From the properties of the exponential function, we know that:
• By variable renaming , and knowing that , it comes that: ( )
=
• The limit of at is:
.
• Eventually, by the computational rules of the limits, it comes that: ( )
=

### Compared growth : basic properties

The exercise deals with the basic rules of "compared growth" between on one hand logarithms or exponentials of a given variable and on the other hand powers of this variable.

• The sentence « » is:
• The sentence « » is .
The true sentence is: « ».
• Formally, this means that: =

### Indeterminate forms with ln or exp

Let be the function defined over by: .

So we have where, for any real in ,   and   .

The aim of the exercise is to evaluate step by step the limit of at .

• Evaluate the limit of at :
=
• The limit of at is:
• Evaluate the limit of at :
=
• The limit of at is:
• Evaluate the limit of at =
• By variable renaming , knowing that , it comes that:
=
• The limit of at is:
.
• Can we deduce the limit at of by applying the computational rules of limits ?
• The computational rules of limits are valid, because there is no indeterminate form. The computational rules of limits are not valid, because there is an indeterminate form .
We use instead the properties of "compared growth": the exponential function dominates any polynom function any polynom function dominates the logarithm function . .
Then it comes:
=

### Basic limits (QUIZZ)

This exercise aims to test the knowledge of basic limits of logarithms and exponentials. Reply as quickly as possible !

 = = = = = =

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• Description: practising with computational rules of limits and indeterminate forms. serveur web interactif avec des cours en ligne, des exercices interactifs en sciences et langues pour l'enseigment primaire, secondaire et universitaire, des calculatrices et traceurs en ligne
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