# Quick limit question

1. May 12, 2010

### 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.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. May 12, 2010

### System

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

3. May 12, 2010

### srfriggen

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

Is that enough proof to show it goes to zero?

4. May 12, 2010

### 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 ..

5. May 12, 2010

### srfriggen

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?

6. May 12, 2010

### srfriggen

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

7. Jun 3, 2010

### System

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