Percent Growth of Exponential Function

[3.141592654]

Homework Statement

If there is a quantity T comprised of two other quantities such that T=L+B, and quantity T and B are both increasing in every period at a fixed percent such that %growth T > %growth B, it will be true that %growth L > %growth T. It will also be true that as n approaches infinity, %growth L will approach %growth T. I am trying to show this.

Homework Equations

The Attempt at a Solution

T is the sum of two other quantities:

$$T=L+B$$

In each period, T grows by the fixed percent g, while B grows by the fixed percent i, and L grows by a percent such that the first equation is true:

$$n=[0,\infty]$$

$$T_{0}(1+g)^n=L_{0}(1+X)^n+B_{0}(1+i)^n$$

To take an example, if g=.1 and i=.05, then T will grow at 10% each year while B will grow at 5% each year. Thus, L must grow at a rate X%>10%. However, as n approaches infinity, X will approach .1. I want to find X as a function of the other variables and show that it approaches g as n approaches infinity:

$$(1+X)^n=\frac{T_{0}(1+g)^n-B_{0}(1+i)^n}{L_{0}}$$

$$n ln(1+X) = ln\left(\frac{T_{0}(1+g)^n-B_{0}(1+i)^n}{L_{0}}\right)$$

$$ln(1+X) = \left(\frac{1}{n}\right)ln\left(\frac{T_{0}(1+g)^n-B_{0}(1+i)^n}{L_{0}}\right)$$

$$e^(ln(1+X)) = e^[(\frac{1}{n})ln(\frac{T_{0}(1+g)^n-B_{0}(1+i)^n}{L_{0}})]$$

$$(1+X) = (e^[ln(\frac{T_{0}(1+g)^n-B_{0}(1+i)^n}{L_{0}})])^\frac{1}{n}$$

$$(1+X) = [\frac{T_{0}(1+g)^n-B_{0}(1+i)^n}{L_{0}}]^\frac{1}{n}$$


I can't get the latex to format so that the quantities I need are being displayed as exponents, but nevertheless, I end up with:

$$X = [\frac{T_{0}(1+g)^n-B_{0}(1+i)^n}{L_{0}}]^(\frac{1}{n})-1$$

Where the fraction 1/n is in the exponent. So I think I've found X, but

$$lim X_{X\rightarrow \infty} = [\frac{T_{0}(1+g)^(\infty)-B_{0}(1+i)^(\infty)}{L_{0}}]^(\frac{1}{\infty})-1$$

The 1/infinity on the right hand side of the above equation is suppose to be an exponent. Also, the numerator of the right hand side is:

To(1+g)^infinity - Bo(1+i)^infinity

I'm not sure how to find this as an indeterminate form.

 

[3.141592654]

I made a notation mistake in my original post. The limit should be as n approaches infinity. So it'd be:

$$lim (X)_{n\rightarrow \infty} = [\frac{T_{0}(1+g)^(\infty)-B_{0}(1+i)^(\infty)}{L_{0}}]^(\frac{1}{\infty})-1$$

However, I'm still stuck at this point.