# Formal definition of limits as x approaches infinity used to prove a limit

1. Oct 15, 2012

### aegiuscutter

1. The problem statement, all variables and given/known data
use the formal definition to show that lim as t goes to infinity of (1-2t-3t^2)/(3+4t+5t^2) = -3/5

2. Relevant equations

given epsilon > 0, we want to find N such that if x>N then absolute value of ((1-2t-3t^2)/(3+4t+5t^2) + 3/5) < epsilon

3. The attempt at a solution
i assume that X>N>0 and that the numerator and denominator can't be equal to zero.
do i have to limit the domain? not sure how to proceed from here

absolute value of ((1-2t-3t^2)/(3+4t+5t^2)) < epsilon - 3/5
absolute value of (-(3t+1)(t-1)/5t^2+4t+3)) < epsilon -3/5

2. Oct 15, 2012

### Zondrina

For this problem, begin by massaging your |f(x) - L| into the form x>N so that you will get a particular value of N which may work.

Then re-state your definition except say $\forall ε>0, \exists N = something \space | \space x > something \Rightarrow |f(x)-L| < ε$

Then proceed to prove that the particular value of N you found satisfies the definition.