Finding the limit of $frac{x}{e^x}$ from the graph of reciprocal function
The problem asks (teachers) to suggest an activity of three questions to students based on this picture to show that $limlimits_{x to +infty} dfrac{x}{e^x}$
(picture : part of the hyperbola associated to $x longmapsto frac 1x$)
The picture suggests working with the following definition of the natural logarithm : $ln(a) := displaystyleint_1^a dfrac 1t dt$.
I considered showing first that $limlimits_{t to +infty} frac{ln t}{t} = 0$ then deduce the desired limit by a change of variable.
The problem is the following :
I first considered upbounding the area below the dark-shaded area by the area of the right shaded triangle but this gives the loose inequality $0<ln(a)<frac a2$ which isn't quite useful to conclude.
I thought about concavity using the equation of the chord joining points $(1,1)$ and $left(a,frac 1a right)$ then integrated the concavity inequality to get :
$ln(a) le frac 12left( a - frac 1aright)$ which doesn't give a proper squeezing argument to conclude for $a>1$ since we only get :
$$0<frac{ln(a)}{a} < frac 12 left( 1 - frac{1}{a^2}right)$$
Thanks in advance for any clues, suggestions.
limits logarithms exponential-function
asked Nov 24 at 4:09
Oussama Sarih
46827
add a comment |
The problem asks (teachers) to suggest an activity of three questions to students based on this picture to show that $limlimits_{x to +infty} dfrac{x}{e^x}$
(picture : part of the hyperbola associated to $x longmapsto frac 1x$)
The picture suggests working with the following definition of the natural logarithm : $ln(a) := displaystyleint_1^a dfrac 1t dt$.
I considered showing first that $limlimits_{t to +infty} frac{ln t}{t} = 0$ then deduce the desired limit by a change of variable.
The problem is the following :
I first considered upbounding the area below the dark-shaded area by the area of the right shaded triangle but this gives the loose inequality $0<ln(a)<frac a2$ which isn't quite useful to conclude.
I thought about concavity using the equation of the chord joining points $(1,1)$ and $left(a,frac 1a right)$ then integrated the concavity inequality to get :
$ln(a) le frac 12left( a - frac 1aright)$ which doesn't give a proper squeezing argument to conclude for $a>1$ since we only get :
$$0<frac{ln(a)}{a} < frac 12 left( 1 - frac{1}{a^2}right)$$
Thanks in advance for any clues, suggestions.
limits logarithms exponential-function
asked Nov 24 at 4:09
Oussama Sarih
46827
1
Have you discussed the Riemann integral in class? I suppose you could break the area into many trapezoids but at a certain point you are just basically doing an integral.
– Mason
Nov 24 at 14:58
add a comment |
The problem asks (teachers) to suggest an activity of three questions to students based on this picture to show that $limlimits_{x to +infty} dfrac{x}{e^x}$
(picture : part of the hyperbola associated to $x longmapsto frac 1x$)
The picture suggests working with the following definition of the natural logarithm : $ln(a) := displaystyleint_1^a dfrac 1t dt$.
I considered showing first that $limlimits_{t to +infty} frac{ln t}{t} = 0$ then deduce the desired limit by a change of variable.
The problem is the following :
I first considered upbounding the area below the dark-shaded area by the area of the right shaded triangle but this gives the loose inequality $0<ln(a)<frac a2$ which isn't quite useful to conclude.
I thought about concavity using the equation of the chord joining points $(1,1)$ and $left(a,frac 1a right)$ then integrated the concavity inequality to get :
$ln(a) le frac 12left( a - frac 1aright)$ which doesn't give a proper squeezing argument to conclude for $a>1$ since we only get :
$$0<frac{ln(a)}{a} < frac 12 left( 1 - frac{1}{a^2}right)$$
Thanks in advance for any clues, suggestions.
limits logarithms exponential-function
asked Nov 24 at 4:09
Oussama Sarih
46827
The problem asks (teachers) to suggest an activity of three questions to students based on this picture to show that $limlimits_{x to +infty} dfrac{x}{e^x}$
(picture : part of the hyperbola associated to $x longmapsto frac 1x$)
The picture suggests working with the following definition of the natural logarithm : $ln(a) := displaystyleint_1^a dfrac 1t dt$.
I considered showing first that $limlimits_{t to +infty} frac{ln t}{t} = 0$ then deduce the desired limit by a change of variable.
The problem is the following :
I first considered upbounding the area below the dark-shaded area by the area of the right shaded triangle but this gives the loose inequality $0<ln(a)<frac a2$ which isn't quite useful to conclude.
I thought about concavity using the equation of the chord joining points $(1,1)$ and $left(a,frac 1a right)$ then integrated the concavity inequality to get :
$ln(a) le frac 12left( a - frac 1aright)$ which doesn't give a proper squeezing argument to conclude for $a>1$ since we only get :
$$0<frac{ln(a)}{a} < frac 12 left( 1 - frac{1}{a^2}right)$$
Thanks in advance for any clues, suggestions.
limits logarithms exponential-function
limits logarithms exponential-function
asked Nov 24 at 4:09
Oussama Sarih
46827
asked Nov 24 at 4:09
Oussama Sarih
46827
asked Nov 24 at 4:09
Oussama Sarih
46827
asked Nov 24 at 4:09
Oussama Sarih
46827
asked Nov 24 at 4:09
Oussama Sarih
46827
46827
1
Have you discussed the Riemann integral in class? I suppose you could break the area into many trapezoids but at a certain point you are just basically doing an integral.
– Mason
Nov 24 at 14:58
add a comment |
1
Have you discussed the Riemann integral in class? I suppose you could break the area into many trapezoids but at a certain point you are just basically doing an integral.
– Mason
Nov 24 at 14:58
1
1
Have you discussed the Riemann integral in class? I suppose you could break the area into many trapezoids but at a certain point you are just basically doing an integral.
– Mason
Nov 24 at 14:58
Have you discussed the Riemann integral in class? I suppose you could break the area into many trapezoids but at a certain point you are just basically doing an integral.
– Mason
Nov 24 at 14:58
add a comment |
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1
Have you discussed the Riemann integral in class? I suppose you could break the area into many trapezoids but at a certain point you are just basically doing an integral.
– Mason
Nov 24 at 14:58