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.
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".
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 thatwhenever
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
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.