# 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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