Thread: Is limit of function approaching infinity still considered to be existent?

1. Is limit of function approaching infinity still considered to be existent?

It is known that $\displaystyle f'(x)=\lim\limits_{t\to x} \frac{f(t)-f(x)}{t-x}$ and we say that $\displaystyle f$ is differentiable at $\displaystyle x$ if this limit exists. But if this limit is $\displaystyle +\infty$ or $\displaystyle -\infty$ as $\displaystyle t\rightarrow x$, can we still say that $\displaystyle f$ is differentiable at $\displaystyle x$, or more precisely, is this limit still considered to exist? Thanks.

2. Originally Posted by zzzhhh
It is known that $\displaystyle f'(x)=\lim\limits_{t\to x} \frac{f(t)-f(x)}{t-x}$ and we say that $\displaystyle f$ is differentiable at $\displaystyle x$ if this limit exists. But if this limit is $\displaystyle +\infty$ or $\displaystyle -\infty$ as $\displaystyle t\rightarrow x$, can we still say that $\displaystyle f$ is differentiable at $\displaystyle x$, or more precisely, is this limit still considered to exist? Thanks.
Infinity is not a number, so the limit does not exist.

(Alternatively why not try constructing an epsilon-delta proof that infinity is a limit)

CB

3. thank you but $\displaystyle +\infty$ or $\displaystyle -\infty$ is a number of the extended real number system, do we have to confine ourselves to the traditional set of reals $\displaystyle \mathbb R$ when working with limits of functions in analysis?

4. Originally Posted by zzzhhh
thank you but $\displaystyle +\infty$ or $\displaystyle -\infty$ is a number of the extended real number system, do we have to confine ourselves to the traditional set of reals $\displaystyle \mathbb R$ when working with limits of functions in analysis?
Look at the altenative suggestion in my original post (though that also is paculiar to the reals).

Also; you are now changing the question, by default in real analysis we are talking about the real, if now we are talking about some version of the extended reals we have a different question.

CB

5. Thank you for your reply. I'm reading Rudin's "Principles of Mathematical Analysis", this book introduces infinite limits(P98) just before the definition of derivatives(P103), so I naturally have the question if infinite limit is regarded as "exists" or not. According to your reply, it is not, so I concluded that we have to confine ourselves to traditional real set in dealing with limits of functions in the definition of derivatives, but I'm not sure, that's my second post. I have refered to other books such as "Mathematical Analysis" written by Apostol, but all these books do not explain explicitly whether infinite limit is allowed. Could you please cite some textbooks stating explicitly that, in the definition of derivatives, infinite limit means the limit does not exist? Thanks!
ps: I have not study real analysis yet. I plan to study it after I finish baby Rudin, and to use Rudin's "Real and Complex Analysis" as textbook, is this book good for self-study?

6. Originally Posted by zzzhhh
Thank you for your reply. I'm reading Rudin's "Principles of Mathematical Analysis", this book introduces infinite limits(P98) just before the definition of derivatives(P103), so I naturally have the question if infinite limit is regarded as "exists" or not. According to your reply, it is not, so I concluded that we have to confine ourselves to traditional real set in dealing with limits of functions in the definition of derivatives, but I'm not sure, that's my second post. I have refered to other books such as "Mathematical Analysis" written by Apostol, but all these books do not explain explicitly whether infinite limit is allowed. Could you please cite some textbooks stating explicitly that, in the definition of derivatives, infinite limit means the limit does not exist? Thanks!
ps: I have not study real analysis yet. I plan to study it after I finish baby Rudin, and to use Rudin's "Real and Complex Analysis" as textbook, is this book good for self-study?
"The limit is infinite" is short hand for "The process diverges" or rather the "limit of the reciprocal is 0".

It is a traditional abuse of language.

CB

7. Many textbooks, and I believe Rudin's is one of them, would say the "$\displaystyle x_n$ diverges to infinity", not "$\displaystyle x_n$ converges to infinity".

8. I finally find an explict answer from wiki. In brief, the limit of the difference quotient exists if the quotient can be "continuouslized" by supplementing a value at zero. Since the codomain of continuous function is only the real set, $\displaystyle \pm\infty$ can not be used as a value for the difference quotient. So, infinite limit means being not differentiable. In addition, the wiki webpage gives an explicit statement that functions with vertical tangent at some point, that is, the limit is $\displaystyle \pm\infty$, is not differentiable there.
Thank you all.

9. Originally Posted by zzzhhh
It is known that $\displaystyle f'(x)=\lim\limits_{t\to x} \frac{f(t)-f(x)}{t-x}$ and we say that $\displaystyle f$ is differentiable at $\displaystyle x$ if this limit exists. But if this limit is $\displaystyle +\infty$ or $\displaystyle -\infty$ as $\displaystyle t\rightarrow x$, can we still say that $\displaystyle f$ is differentiable at $\displaystyle x$, or more precisely, is this limit still considered to exist? Thanks.
Go back to the definition. Let $\displaystyle f$ be a function defined in a neighborhood of $\displaystyle a$. We say the limit of $\displaystyle f$ at $\displaystyle a$ exists iff there exists a real number $\displaystyle L$ such that for any $\displaystyle \varepsilon > 0$ there is a $\displaystyle \delta>0$ such that $\displaystyle 0<|x-a|<\delta \text{ and }x\in \text{dom}(f) \implies |f(x)-L|<\epsilon$. If $\displaystyle \lim_{x\to a}f(x)=\infty$ then it is impossible to satisfy that condition above, so there is no limit.

10. Originally Posted by ThePerfectHacker
... then it is impossible to satisfy that condition above, so there is no limit.
Beautifully stated.

11. Although my question is solved, I'd like to say something about the existence of infinite limit. when I studied advanced calculus, I often met notations such as $\displaystyle \lim\limits_{t\to+\infty}, \lim f(x)\rightarrow+\infty$, or $\displaystyle \smallint \nolimits_a^{+\infty}...$. At first I just cannot understand what the symbol "$\displaystyle \pm\infty$" actually is, of course it is not a real number, but it occurs frequently in the textbook, with some strange properties such as $\displaystyle a+(+\infty)=+\infty,1/+\infty=0$. Sometimes I have to write $\displaystyle x\rightarrow+\infty$ by hand but says "Oh, it diverges" in heart. This antithesis lasts for years until I studies the general topology, especially the notion of compactification. There, "$\displaystyle \pm\infty$" together with real set $\displaystyle \mathbb R$ constutute a compact space, $\displaystyle +\infty$ is just a common element which is nothing different from real numbers, and $\displaystyle x\rightarrow+\infty$ or $\displaystyle \lim f(x)=+\infty$ which means divergence in advanced calculus IS indeed a convergence in topology. In topology the convergence means there is always a $\displaystyle \delta$-neighborhood of the variable such that the values can all fall into any $\displaystyle \epsilon$-neighborhood. To be more precise in limits of functions (assuming defined everywhere), $\displaystyle f$ converges to $\displaystyle a$ at $\displaystyle x$ iff for any neighborhood $\displaystyle B(a,\epsilon)$ of $\displaystyle a$(here I use basis element instead), there is a neighborhood of $\displaystyle x$ excluding $\displaystyle x$, that is, $\displaystyle B(x,\delta)-\{x\}$, such that $\displaystyle f(t)\in B(a,\epsilon)$ whenever $\displaystyle t\in B(x,\delta)-\{x\}$. Since $\displaystyle \{\pm\infty\}\cup\mathbb R$ is an order topology, the neighborhood of $\displaystyle +\infty$ is just $\displaystyle (x,+\infty]$ where $\displaystyle x$ is any real, so the traditional convergence is of course convergence, but for $\displaystyle \lim\limits_{x\to a}f(x)=+\infty$, that is, for any real $\displaystyle M$, no matter how large it may be ,there is a $\displaystyle x$ in a $\displaystyle \delta$-neighborhood of $\displaystyle a$ such that $\displaystyle f(x)>M$. But look at what $\displaystyle f(x)>M$ means? it just means $\displaystyle f(x)\in (M,+\infty]$, a neighborhood of $\displaystyle +\infty$! so according to the aforementioned definition of convergence in topology, $\displaystyle f$ DOES CONVERGE to $\displaystyle +\infty$, NOT diverge. So it is not an abuse of the word "limit" or "converge", nor is the limit inexistent. As for my question, although it is often confused if $\displaystyle \pm\infty$ is allowed in a notion, the codomain of a function is definite, we are always working with functions in $\displaystyle \mathbb R$, i.e., the codomain is $\displaystyle \mathbb R$, not including $\displaystyle \pm\infty$; there is no confusion regarding this. So in the definition of derivative, using the "continuouslization" of the difference quotient, we can conclude that infinite limit means the derivative does not exist, although the function limit does exist -- =$\displaystyle +\infty$. As a result, textbooks should not only says the limit exists, but also should add a note that the limit not be $\displaystyle \pm\infty$ in defining derivatives. To sum up, the question is not whether the limit exist, but whether we should confine or restrict ourselves to real set only when understanding this concept.
Since we have learned topology, why not hold a unified point of view, although sometimes at the cost of confusion whether $\displaystyle \pm\infty$ is included or not.