How to Calculate the Limit of 4^x/7^x+4 as x Approaches Infinity?

[srfriggen]

how do you calculate lim x-> inf 4^x/7^x+4

It seems obvious that the bottom is growing at a faster rate than the top, so the limit would be zero, but how would you show that algebraically? I tried using l'hopital and just wind up with:

4^x ln4 / 7^x ln7

and it doesn't seem that using l'hopital again and again would get me anywhere further.

 

[System]

Personally, I would divide top and bottom by 7^x.

 

[srfriggen]

Personally, I would divide top and bottom by 7^x.

ok so that leaves lim x->inf (4/7)^x

Is that enough proof to show it goes to zero?

 

[System]

if |a|<1 then lim(x-->infinity) for a^x = 0
If you see the graph for any function a^x where |a|<1, you see this is true
You can prove that by using the convergence's tests for the infinite series ..

 

[srfriggen]

if |a|<1 then lim(x-->infinity) for a^x = 0
If you see the graph for any function a^x where |a|<1, you see this is true
You can prove that by using the convergence's tests for the infinite series ..

ah ok,

so that would be a geometric series with l r l < 1 (which we know converges), or you can prove this converges by using the root test, 4/7 < 1 , so since the series converges the limit of the sequence or partial sums goes to zero?

do I have that all correct?

 

[srfriggen]

sorry, I meant the limit of the nth partial sum goes to zero as n goes to infinity

 

[System]

No.
The limit of the nth term is zero.
Not the limit of the sequence of the partial sums!