Thread: 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.

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.

Originally Posted by VonNemo19

Well, share the question with us, and maybe we can help.

Oops... my bad.

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".

Originally Posted by nein12

Oops... my bad.

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 and as ? ?, then

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

if for every number M>0, there is some number such that

whenever

So then the proof is straight forward

if for every x arbitrarily close to a, then for every value of x arbitrarily close to a

Therefore

whenever

Originally Posted by VonNemo19

So we wish to show that if and as ? ?, then

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

if for every number M>0, there is some number such that

whenever

So then the proof is straight forward

if for every x arbitrarily close to a, then for every value of x arbitrarily close to a

Therefore

whenever

Umm... I am very good with math. What is M and delta mean?
And I've never seen expression before, though I kinda understand it.

Originally Posted by nein12

Umm... I am very good with math. What is M and delta mean?
And I've never seen 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.