Help me to calculate this limit

[-TesseracT-]

How do you calculate the limit of (e^(3x) - 1) / (e^(4x) - 1) as x -> 0 ?

 

Thanks.

[8uhbbhu8]

(If I'm reading this right (it's real late for me )

 

First plug in the given x 

 

Then Simplify

 

 

What is 3*x

What is 4*x

 

So now your down to this

 

(e^0-1)/(e^0-1)

 

Use that as a start then follow your basic math rules. But I almost feel like its missing something....

 

I'm tempted to say try to factor it but it's been too long since I did any of that

[rtournem]

Hello!

You can factor in order to remove the (1-1) problem using the "identités remarquables" (I'm french ^^):

 

(e^(3x) - 1) / (e^(4x) - 1)=A/B

 

A=(e^x)^3 - 1^3)=(e^x-1)(e^(2x)+e^x+1)

 

B=((e^(2x))^2 - 1^2)=(e^(2x)-1)(e^(2x)+1)=(e^(x)-1)(e^x+1)(e^(2x)+1)

 

You've done the hardest. Now just simplify (e^x-1):

 

A/B=(e^(2x)+e^x+1)/(e^x+1)(e^(2x)+1)  --- x=0 ---> 3/4

 

Tadammmmm!

 

If you hate to have to "see" the solution just use the "développement limité" around 0 of f(x) (In your case f(x)=e^(4x) and f(x)=e^(3x))

https://fr.wikipedia.org/wiki/Développement_limité

 

 

[-TesseracT-]

3 hours ago, 8uhbbhu8 said:

(If I'm reading this right (it's real late for me )

 

First plug in the given x 

 

Then Simplify

 

 

What is 3*x

What is 4*x

 

So now your down to this

 

(e^0-1)/(e^0-1)

 

Use that as a start then follow your basic math rules. But I almost feel like its missing something....

 

I'm tempted to say try to factor it but it's been too long since I did any of that

But (e^0-1)/(e^0-1) = 0/0, so that doesnt work. thanks for reply anyways.

 

7 minutes ago, rtournem said:

Hello!

You can factor in order to remove the (1-1) problem using the "identités remarquables" (I'm french ^^):

 

(e^(3x) - 1) / (e^(4x) - 1)=A/B

 

A=(e^x)^3 - 1^3)=(e^x-1)(e^(2x)+e^x+1)

 

B=((e^(2x))^2 - 1^2)=(e^(2x)-1)(e^(2x)+1)=(e^(x)-1)(e^x+1)(e^(2x)+1)

 

You've done the hardest. Now just simplify (e^x-1):

 

A/B=(e^(2x)+e^x+1)/(e^x+1)(e^(2x)+1)  --- x=0 ---> 3/4

 

Tadammmmm!

 

If you hate to have to "see" the solution just use the "développement limité" around 0 of f(x) (In your case f(x)=e^(4x) and f(x)=e^(3x))

https://fr.wikipedia.org/wiki/Développement_limité

 

 

Wow, that's a brilliant solution, thanks a lot!

[You_are_a_cunt]

Dammit I started on this and then a wild coffee break appeared.

But yeah, @rtournem's solution is accurate

[Dash Lambda]

I do it the calculus way:

lim_x→0 f(x)/g(x) = lim_x→0 f'(x)/g'(x)

lim_x→0 (e^3x-1)/(e^4x-1) = lim_x→0 (3e^3x)/(4e^4x) = 3/4