Solve Indeterminate Form: lim_n\rightarrow\infty A^n-B^n

[3.141592654]

Homework Statement

Find lim_n\rightarrow\infty A^n-B^n

Homework Equations

The Attempt at a Solution

This leads to the indeterminate form:

lim_n\rightarrow\infty A^n-B^n = A^\infty-B^\infty = \infty-\infty

I've tried to find a common denominator that would simplify this limit but haven't had any luck. Is it possible to determine this limit? If so, can anyone point me in the right direction to begin solving this?

Thanks.

 

[3.141592654]

p.s. I am unable to edit LaTex for some reason. Is anyone else experiencing this issue?

 

[Mark44]

I fixed your LaTeX.

Homework Statement

Find lim_{n\rightarrow\infty} A^n-B^n

Homework Equations

The Attempt at a Solution

This leads to the indeterminate form:

lim_{n\rightarrow\infty} A^n-B^n = A^{\infty}-B^{\infty} = \infty-\infty

You can never substitute infinity in as if it were a value. Without knowing more about A and B, you can't evaluate the limit. IOW, there's nothing more you can do with this:
lim_{n\rightarrow\infty} A^n-B^n

I've tried to find a common denominator that would simplify this limit but haven't had any luck. Is it possible to determine this limit? If so, can anyone point me in the right direction to begin solving this?

If you were trying see what you did using Preview Post and making changes, this is a known problem. If you refresh your browser you will see the changes you made. Otherwise, what you see is what was most recently in the browser cache.

 

[3.141592654]

Thanks for the reply.

I suppose that for some context, I should tell you that I'm trying to show that if you have a quantity T that grows by a constant percent each year, and T=L+B, with B also growing by a constant percent each year, if T is growing faster than B, then L will grow by a percent greater than T. Also, as time approaches infinity, the percent that L grows at will approach the percent that T grows at.

A and B were just values I substituted because I was trying to make the problem easier. So I started with this:

T(1+g)^n=L(1+X)^n+B(1+i)^n

Where T=L+B, g>i

I solved for X and got:

X(n)=(\frac{T(1+g)^n-B(1+i)^n}{L})^\frac{1}{n}-1

And now I'm trying to show that as n approaches infinity that X will approach g. However, I'm at a loss as to how to solve it:

lim_{n\rightarrow\infty} (\frac{T(1+g)^n-B(1+i)^n}{L})^\frac{1}{n}-1

I thought that I could evaluate the inner function first and then move on to develop an answer in steps, but this can't be done as I originally thought. Is there any way to solve this, and how can I get started?

 

[Mark44]

You might be able to replace each of the expressions (1 + <whatever>)ⁿ by a finite binomial series (see http://en.wikipedia.org/wiki/Binomial_series).