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.
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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