# Math Help - Proving the limit...

1. ## Proving the limit...

I get how to get the answer, but I need to use formal definition of limits... something I am very unfamiliar with.

lim x->a g(x) = infinity and g(x) < or equal f(x) for x -> a

I know the answer is lim as x->a for f(x) = infinity... but what definitions do I need to do this? I am not sure how to show work for this question.

2. Originally Posted by nein12
I get how to get the answer, but I need to use formal definition of limits... something I am very unfamiliar with.

lim x->a g(x) = infinity and g(x) < or equal f(x) for x -> a

I know the answer is lim as x->a for f(x) = infinity... but what definitions do I need to do this? I am not sure how to show work for this question.
Well, share the question with us, and maybe we can help.

3. Originally Posted by VonNemo19
Well, share the question with us, and maybe we can help.

If lim x->a g(x) = infinity and g(x) < or equal to f(x) for x -> a, then limx-> a f(x) = infinity

Prove, using the formal definition of limits, each of the statement is true.

So... that's the question. Honestly, I never had to answer this kind of questions. It's usually applications format or equation format.

Another one is limx->infinity f(x) = -infinuty and c > 0, then limx ->infinity c f(x) = - infinity

Not sure which rules to use, but I can reason... that f(x) is a negative infinity... times bu any constant positive number is still negative infinity. I am not sure if there are any "specific rules".

4. Originally Posted by nein12

If lim x->a g(x) = infinity and g(x) < or equal to f(x) for x -> a, then limx-> a f(x) = infinity

Prove, using the formal definition of limits, each of the statement is true.

So... that's the question. Honestly, I never had to answer this kind of questions. It's usually applications format or equation format.

Another one is limx->infinity f(x) = -infinuty and c > 0, then limx ->infinity c f(x) = - infinity

Not sure which rules to use, but I can reason... that f(x) is a negative infinity... times bu any constant positive number is still negative infinity. I am not sure if there are any "specific rules".
So we wish to show that if $\lim_{x\to{a}}g(x)=\infty$ and $g(x)\leq{f(x)}$ as ? $x\to{a}$ ?, then $\lim_{x\to{a}}f(x)=\infty$

Well, for infinite limits the definition states:
Let g(x) be a function that is defined on an interval containing a, except possibly at a. Then we say that

$\lim_{x\to{a}}g(x)=\infty$

if for every number M>0, there is some number $\delta>0$ such that
$g(x)>M$ whenever $0<\mid{x-a}\mid<\delta$

So then the proof is straight forward

if $g(x)>M$ for every x arbitrarily close to a, then $f(x)>g(x)$ for every value of x arbitrarily close to a $\Rightarrow{f(x)>M}$

Therefore
$f(x)>M$ whenever $0<\mid{x-a}\mid<\delta$

5. Originally Posted by VonNemo19
So we wish to show that if $\lim_{x\to{a}}g(x)=\infty$ and $g(x)\leq{f(x)}$ as ? $x\to{a}$ ?, then $\lim_{x\to{a}}f(x)=\infty$

Well, for infinite limits the definition states:
Let g(x) be a function that is defined on an interval containing a, except possibly at a. Then we say that

$\lim_{x\to{a}}g(x)=\infty$

if for every number M>0, there is some number $\delta>0$ such that
$g(x)>M$ whenever $0<\mid{x-a}\mid<\delta$

So then the proof is straight forward

if $g(x)>M$ for every x arbitrarily close to a, then $f(x)>g(x)$ for every value of x arbitrarily close to a $\Rightarrow{f(x)>M}$

Therefore
$f(x)>M$ whenever $0<\mid{x-a}\mid<\delta$
Umm... I am very good with math. What is M and delta mean?
And I've never seen $0<\mid{x-a}\mid<\delta$ expression before, though I kinda understand it.

6. Originally Posted by nein12
Umm... I am very good with math. What is M and delta mean?
And I've never seen $0<\mid{x-a}\mid<\delta$ expression before, though I kinda understand it.

The problem you posted requires knowlege of the epsilon-delta (or formal) defintion of limit.

M is some arbitrarily large number, loosely speaking, delta is the distance x is away from a, and the inequality is a way of narrowowing down the distance between a and x to any desired length.