Formulate Precise Definition of a Limit

[nycmathguy]

Calculus
Section 2.6

We end Section 6 here.

 

[MathLover1]

formulate precise definition of lim(f(x)=-∞ then use definition to prove that lim(1+x^3)=-∞

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->-∞

 

[nycmathguy]

formulate precise definition of lim(f(x)=-∞ then use definition to prove that lim(1+x^3)=-∞

View attachment 3332
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->-∞

OMG!!! This is a monster problem. Thank you.