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

[aegiuscutter]

Homework Statement

use the formal definition to show that lim as t goes to infinity of (1-2t-3t^2)/(3+4t+5t^2) = -3/5

Homework 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

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

 

[STEMucator]

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&gt;0, \exists N = something \space | \space x &gt; something \Rightarrow |f(x)-L| &lt; ε

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