I have to proof that $\lim_{x \to \c} frac{1}{f(x) = 0$

[kasperrepsak]

Homework Statement

Given is: f is a function that maps D onto the Real numbers, and c is within D and it is a limit point, and f(x) =/= 0 for all x in D, and $\lim_{x \to c} f(x) = \infty$
I have to proof that:
$\lim_{x \to c} \frac{1}{f(x)} = 0$

The Attempt at a Solution

This means that according to the definition I have to proof that $\forall \epsilon \ \exists \delta$ so that $\forall x \in D \ with \ 0 < |x-c|< \delta \ \ : |\frac{1}{f(x)} - 0|< \epsilon$.

Im not sure how to go on from here. Or do i have to do something else?

 

[jbunniii]

Hint:
$$\left|\frac{1}{f(x)} - 0\right| < \epsilon$$
is equivalent to
$$|f(x)| > \frac{1}{\epsilon}$$

 

[kasperrepsak]

Would it then be enough to say that because f diverges to infinity as x approaches c, this by definition means that there must be a $\delta$ for which if $0<|x-c|<\delta$ the latter inequality that you wrote holds, which is equivalent to the former, quod erat demonstrandum?

 

[SammyS]

Homework Statement

Given is: f is a function that maps D onto the Real numbers, and c is within D and it is a limit point, and f(x) =/= 0 for all x in D, and $\lim_{x \to c} f(x) = \infty$
I have to proof that:
$\lim_{x \to c} \frac{1}{f(x)} = 0$

The Attempt at a Solution

What you have to prove is what you have in the following statement .

This means that according to the definition I have to proof that $\forall \epsilon \ \exists \delta$ so that $\forall x \in D \ with \ 0 < |x-c|< \delta \ \ : |\frac{1}{f(x)} - 0|< \epsilon$.

I'm not sure how to go on from here. Or do i have to do something else?

What will help you get there is to state what is meant by:

$\displaystyle \lim_{x \to c} f(x) = \infty\ .$

 

[kasperrepsak]

Well the weird thing is that in our book we hv only defined what is meant by a sequence diverging to infinity at a point c. This assignment is part of the chapter on continuity where we've only defined what is meant by a limit of f as x goes to infinity. But wut i wrote should b ok since that follows from the definition right?

 

[SammyS]

Well the weird thing is that in our book we have only defined

Spoiler

∫

what is meant by a sequence diverging to infinity at a point c. This assignment is part of the chapter on continuity where we've only defined what is meant by a limit of f as x goes to infinity. But what i wrote should be ok since that follows from the definition right?

What you wrote is what needs to be proved.

What you need, among other things, to get there is a definition of what it means for $\displaystyle \lim_{x \to c} f(x) = \infty\ .$

You may be able to deduce this from

what is meant by a sequence diverging to infinity at a point c.​

and from

what it means for the limit of a function, f(x), to converge as x → c .​

Basically, what it means for $\displaystyle \lim_{x \to c} f(x) = \infty\,,$ is that given any M > 0, (usually M is a large number) there exists a δ > 0 such that for any x satisfying 0<|x-c|<δ, you have that f(x) > M .

 

[kasperrepsak]

No but i ment what i wrote in reply to the firs post. There must be a delta greater then 1/epsilon

 

[SammyS]

No but i meant what i wrote in reply to the firs post. There must be a delta greater then 1/epsilon

That works, if you spell out why it works.

Given an ε > 0, then 1/ε > 0 . Since $\displaystyle \ \lim_{x \to c} f(x) = \infty\ \$ there exists δ > 0, such that ... f(x)>1/ε ...

Added in Edit:

I removed the absolute value from f(x) above

 

[kasperrepsak]

OK thank you. Yeah I know that in writing a proof one has to write a lot of little things to make it formally correct, I just wanted to know if I understand the method of proofing. Thanks again : )