limits – In this textbook explanation of needing partial derivatives, how is this partial derivative not an indeterminate form?

$$ f(x,y) = x^frac{1}{3}y^frac{1}{3} $$

$$frac{partial f}{partial x}(0,0) = lim_{x to 0} frac{f(h,0)-f(0,0)}{h}= lim_{x to 0} frac{0-0}{h} = 0$$

“and, similarly, $frac{partial f}{partial y}(0,0) =0$ (these are not indeterminate forms!). It is necessary
to use the original definition of partial derivatives, because the functions $x^frac{1}{3}$ and $y^frac{1}{3}$
are not themselves differentiable at 0.”

This is a portion of textbook explaning why a simple definition of a partial derivative does not work but a linear approximation definition of a partial derivative must be used.

However, I’m confused at this part where they seem to be trying to use a counterexample to prove why a simple definition of partial derivatives does not work. Isn’t this limit an indeterminate form? Yet, as you can see, the textbook claims this limit is not an indeterminate form to make their case.

I would greatly appreciate your help in making sense of this textbook.

Reference textbook: Vector Calculus by Marsden and Tromba 5th edition.

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limits – Calculating $lim _{nto infty }left(frac{1cdot n + 2cdot(n-1) + 3cdot (n-2)+ … +1cdot n}{n^2}right)$?

Hello everyone how can I calculate the limit of:

$lim _{nto infty }left(frac{1cdot n + 2cdot(n-1) + 3cdot (n-2)+ … +1cdot n}{n^2}right)$?

My direction was to convert it to something looks like Riemann sum by doing this:

$lim _{nto infty }left(frac{sum_{k=0}^{n} (k+1)(n-k)}{n^2}right)$

But I don’t know how to continue.

celery – Memory limits in systemd

I am using systemd to run Celery, a python distributed task scheduler, on an Ubuntu 18.04 LTS server.

I have to schedule mostly long running tasks (taking several minutes to execute), which, depending on data, may end up eating a large amount of RAM. I want to put on some sort of safety measure in order to keep memory usage under control.

I have read here that you can control how much RAM it’s used by a systemd service using the MemoryHigh and MemoryMax options

I have used these options in my Celery service setup, and I watched what happens to a celery service when it reaches the given limits with htop.

The service stops the execution and is put on “D” state, but stays there and the allocated memory is not freed.

Is it possible for systemd to kill the memory eating process?

double integration u+v substitution selection and sensitivity to limits (astronomy problem)

I’m trying to understand an integration in a paper https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1008988&tag=1 by working through it by hand

formula for perfectly focused point source from Liebe paper

The limits for both integrals are -0.5 to 0.5 representing the grid of pixels on the imager imaging a focused star and if I use Wolfram the answer comes to 0.38 as stated

I used your video as a guide to a hand solution https://www.youtube.com/watch?v=UrTGqJraxXA
I went with u = x^2+y^2 for the first substitution

My problem I’m stuck at the choice for the second substitution v as I can’t use v=1+y^2 because I don’t have a 2 from the xy term to play with (can’t

Can anybody offer any guidance as to choice of u and v

I also note that the result in the case of limits 0 to 1 being used rather than -0.5 to 0.5 the result is 2x 0.38 … again any guidance you can offer would be appreciated

The reason for this is I want to understand the effect of defocusing on centroid determination as the practice in the paper is to deliberately defocus so as to get a 3×3 pixel spot on the imager

Google analytics embed API limits

If I use google Embed APIs to build a custom dashboard in my website, how does the limits work ? I understand it internally uses the google core reporting APIs.
Is there any limits to this API ?

Let’s say I would like to query about 1000 times each with different dimensions/filters.
That would be 1000 embeddings, how does the limits work in this case ? will I have to take account of number of queries made to the core reporting APIs internally ?

windows 10 – How to remove limits on Disk Drive space?

Migrated from Hardware Reccomendations SE

I am looking to upgrade my ASUS ROG GL703GE to have more disk space and memory, and have run into a few constraints laid out by the manufacturer. According to the specification site, the computer supports up to 2TB SATA for an HDD and up to 512GB SATA for SSD. If I’m upgrading my computer, I might as well go all-in, so I’d like to use a 2TB SATA SSD.

I’m assuming that, for whatever solutions exist for the disk, a similar solution can be used to upgrade memory. If not, I’ll ask a separate question for that.

What can I do to remove the limit on the disk size for my device? I’m on Windows 10 x64, so there’s no problems with the size of the disk itself, just the conversion from HDD to SSD.

Show that a function f, continuous on (a,b), has an abs minimum value. The limits as x approaches either bounds of the interval is +Infinity

Can you show that the function f must have an absolute minimum value on the interval (a,b), if f is continuous on (a,b) and the right hand limit as x->a along with the left hand limit as x-> b are both equal to positive infinity?

limits – Proving that pole of rational polynomial “goes to” $+infty$ or $-infty$

Lets define the rational function:

$$f(x) = sumlimits_{k=1}^{infty} frac{x^s}{(1-x^k)^2}, quad s in mathbb{N}_0$$

For me, it is clearly to see that it has two poles: at $x = -1$ and at $x = 1$. At the pole, the denominator has a root, so the pole is in general undefined because it would be a division by zero.

If I look at the plot, I can see how the plot behaves if I near from the right or left side to the pole. But is there a relative easy way (like finding the poles) to show that

$$limlimits_{x to -1}f(x) = begin{cases}+infty & text{if } s text{ is even}\-infty & text{if } s text{ is odd}end{cases}$$

How can I save GoToMeeting on my computer without limits?

What is the best GoToMeeting recorder? I wish to save the conference recording for future reference without using the integrated method. I tried the official method but it was annoying. I want to find an easy to use screen recorder with power functions. Could you recommend one for me? thank you so much