formulate precise definition of lim(f(x)=-∞ then use definition to prove that lim(1+x^3)=-∞
[ATTACH=full]3332[/ATTACH]
Definition: Infinite Limit at Infinity (Formal)
We say a function f has an infinite limit at infinity and write lim(f(x)=infinity as x-> ∞
if for all M>0, there exists an N>0 such that f(x)>M for all x>N (see Figure).
We say a function has a negative infinite limit at infinity and write
limx→∞f(x)=−∞
if for all M<0 , there exists an N>0 such that f(x)<M for all x>N .
Similarly we can define limits as x→−∞.
prove that lim(1+x^3)=-∞
Apply Infinity Property:
lim (x->-∞, (ax^n+..... +bx+c )=-∞ , a>0, n is
in your case a=1 and n=3
proof:
Given ϵ >0, we need δ >0 such that if 0<| 1+x|<δ, then |1+x^3|<ϵ.
Now,
|1+x^3|=|(x + 1) (x^2 - x + 1)|
If |x+1|<1, that is, −1<x+1<1, then note that
−1<x+1<1 <=> -2<x<0 <=> x^2-x+1<0=> 0^2-0+1=1
and so
|1+x^3|=|x+1|(x^2-x+1)<1|x+1|
|1+x^3|=|x+1|(x^2-x+1)<|x+1|
So if we take δ=min(1,ϵ*1), then
0<|x+1|<δ <=> |1+x^3|=|x+1|(x^2-x+1)<ϵ
since ϵ = -∞ in your case we proved that
lim(1+x^3)=-∞ as x->-∞
