common Krait

Cute snake. I hope it's not poisonous.

I guess I can't put link but if you go through my blog you will find the story

cool to look forward to 😅

Part 1/6:

Understanding the Limit of a Vector Function

The concept of limits is fundamental to understanding calculus, not just for real-valued functions but also for vector functions. In this article, we will explore how the limit of a vector function is defined, how it relates to its component functions, and delve into some examples for clearer understanding.

Definition of Limits for Vector Functions

The limit of a vector function ( \mathbf{r}(t) ) can be defined by examining the limits of its individual components. Specifically, if a vector function comprises three component functions ( f(t), g(t), ) and ( h(t) ) in three-dimensional space, then the limit as ( t ) approaches ( a ) can be expressed as follows:

[

\lim_{t \to a} \mathbf{r}(t) = \begin{pmatrix}

Part 2/6:

\lim_{t \to a} f(t) \

\lim_{t \to a} g(t) \

\lim_{t \to a} h(t)

\end{pmatrix}

]

This definition stipulates that the limits of the component functions must exist for the limit of the vector function to be defined. Thus, the limit of the vector function as ( t ) approaches ( a ) results from combining the limits of its components.

Length and Direction

When we state that

[

\lim_{t \to a} \mathbf{r}(t) = \mathbf{l}

]

we can interpret this to mean that as ( t ) approaches ( a ), the length and direction of the vector ( \mathbf{r}(t) ) converge towards those of another vector ( \mathbf{l} ). In essence, the behavior of the vector can be quantitatively described by considering how close it gets to the vector ( \mathbf{l} ) in terms of both magnitude and direction.

Part 3/6:

Visualization of Limits

To visualize this relationship, imagine a three-dimensional coordinate space where we visualize the vector ( \mathbf{l} ) as a line in this space. As ( t ) approaches ( a ), the vector function ( \mathbf{r}(t) ) draws nearer to this line (or vector). This geometric interpretation helps clarify how limit processes in vector calculus can describe trajectories and convergence of vectors.

Properties of Limits

Similar to limits of real-valued functions, limits of vector functions obey analogous rules. For instance, the limit of a sum or scalar multiple of vector functions follows predictable patterns, just like in the one-dimensional case.

Example: Finding the Limit of a Vector Function

Part 4/6:

To solidify our understanding, let’s examine an example where we find the limit of a vector function:

Given:

[

\mathbf{r}(t) = (1+t^3) \mathbf{i} + (t e^{-t}) \mathbf{j} + \left(\frac{\sin t}{t}\right) \mathbf{k}

]

We want to calculate the limit as ( t ) approaches ( 0 ).

Solution:

Applying our earlier definition, we will evaluate the limit of each component:

[

\lim_{t \to 0} \mathbf{r}(t) = \lim_{t \to 0} (1+t^3) \mathbf{i} + \lim_{t \to 0} (t e^{-t}) \mathbf{j} + \lim_{t \to 0} \left(\frac{\sin t}{t}\right) \mathbf{k}

]

Calculating each limit gives:

1. ( \lim_{t \to 0} (1 + t^3) = 1 + 0 = 1 ) for ( \mathbf{i} ) component.

2. ( \lim_{t \to 0} (t e^{-t}) = 0 \cdot 1 = 0 ) for ( \mathbf{j} ) component.

Part 5/6:

1. For the ( \mathbf{k} ) component, ( \frac{\sin t}{t} ) tends to ( 1 ) as ( t ) approaches ( 0 ).

Thus, the overall limit becomes:

[

\lim_{t \to 0} \mathbf{r}(t) = 1 \mathbf{i} + 0 \mathbf{j} + 1 \mathbf{k} = \mathbf{i} + \mathbf{k}

]

In component form, it simplifies to:

[

\begin{pmatrix}

1 \

0 \

1

\end{pmatrix}

]

Recap: L'Hôpital's Rule

In scenarios where limits result in indeterminate forms such as ( \frac{0}{0} ), L'Hôpital's Rule can provide a resolution. If ( f ) and ( g ) are differentiable functions, and the limits ( \lim_{x \to a} f(x) ) and ( \lim_{x \to a} g(x) ) both equal ( 0 ), then we may find:

[

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

]

provided that this latter limit exists.

Conclusion

Part 6/6:

Understanding the limits of vector functions is a crucial component of multivariable calculus, providing insight into their behavior as parameters change. By employing definitions and visualizations, along with practical examples and established rules, we can better grasp the intricacies of vector limits and their applications in more complex mathematical scenarios.

I also don't use Iphone and I never used it. So, I have no use of those code.

I prefer Android phones myself. 😀

Same here! Never touched an iPhone, team Samsung all the way.

Still, these secret codes are kinda cool to know, just in case we ever get curious!

Generous and honest!

Appreciate the share someone out there will be dialing away like a secret agent now.

Cool to see these shortcuts that might make someone's life easier. I never owned an iPhone either, so I will let others enjoy them.

Never owned an iPhone either, but this list is tempting me to borrow one just to try them out!

WOw first time hearing about most of the codes , thanks for sharing

Externalizers may appear virtuous by distancing themselves from guilt and portraying their errors as the fault of others. Yet, this deceptive practice traps them in a mental prison that obstructs authentic, measurable growth in maturity.

Thus, by refusing to acknowledge mistakes, misjudgments, or questionable actions, a person may feel justified in their narrative—but the cost is steep. Avoiding accountability ensures a lifetime of missed opportunities for true advancement.

This perspective isn't about accepting every error but about having the inner resilience to face and own shortcomings. Embracing accountability is a rigorous measure of maturity, reflecting both inherent respectability and personal agency.

Those who shy away from it remain caught in a state of arrested development, burdened by the price of maintaining a flawless self-image at the expense of self-honesty.

In this video I go over the definition of the limit of a vector function, which involves taking the limit of each of its component functions. If the limit exists, then so does the limit of each component function.

Lions to the moon!

#ai

Welcome back, Jongo. I doubt you can stay a day without the checkmark. Haha.

Once a premium, always a premium.

~ Winanda

Welcome back to the premium squad!

@simplegame clearly knows how to bring the lions home.

$LEO at $0.02 is just the ignition—next stop: orbit! 🦁

Sources:

• https://www.reuters.com/technology/microsoft-unit-russia-file-bankruptcy-database-shows-2025-05-30/
• https://telegrafi.com/en/Putin-orders-switch-to-free-software/

1. https://discord.com/channels/484546513507188745/484571108885135361/1378862423062085714

2. 💪 https://x.com/ItaliaDash/status/1929396397599510571

A prominent crypto asset has contributed to staving off financial collapse, yet unchecked government spending suggests that only sweeping productivity improvements can secure the nation's future.

"One observation noted, 'Perhaps shifting

away from politics in favor of leveraging humanoid robots to boost GDP growth is the only rational option if no other solution exists.'"

1. 🇬🇧 https://docs.dash.org/en/stable/docs/user/introduction/about.html
2. 🇮🇹 https://docs.dash.org/it/stable/docs/user/introduction/about.html

Stake based curation parameters

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https://inleo.io/@stekene/discovering-a-new-community-moonkart-racers-hub-4mb

https://inleo.io/@vickoly/small-oversight-big-wakeup-call-5jc

https://inleo.io/@henrietta27/what-i-ordered-vs-what-i-got--hym?referral=henrietta27

https://inleo.io/@wiseagent/testing-my-patience-with-apple-r9

https://inleo.io/@the-african-man/proven-strategies-to-follow-for-a-successful-crypto-trading-bpb?referral=the-african-man

https://inleo.io/@rtonline/last-brawl-report-my-winning-streak-in-brawls-purchasing-and-opening-gladius-cases-splinterlands-fvq

https://inleo.io/@mercurial9/bloodline-dominion-and-shapeshift-splinterlands-446-2hv

https://inleo.io/threads/view/mercurial9/re-leothreads-29xb4gmth

https://inleo.io/@imfarhad/myhivegoals-june-monthly-goals-and-progress-updates-9nu

Check out the original post for more photos from this set: https://ecency.com/photography/@darth-azrael/vintage-photos-lot-3-521

What a beautiful woman. She's probably someone's grandma now.

So many inconsiderate people like that on our side of the world. Playing music loudly, eating smelly food in public transport, etc.

Src: https://www.tumblr.com/70sscifiart/784625657447677952/me-logging-on-to-work-after-the-long-weekend?source=share

Even when the waves say "no swim today," the sand still whispers "come play!"

Beach strolls, laughter, and chilly toes, nature’s way of saying fun has no rules.

A stroll along the shoreline, good company, salty breeze sometimes that's all the fun you need.

Interesting and I spot some motorcycles in the back. I like that the beach looks quite wild, probably a lot of tranquility around it... or not?

Even when the sea feels like ice the beach still welcomes you.

Walking is refreshing on beaches only if the breeze is not too cold. Good morning

A good way to have fun with family and friends. Good morning

So true! Even if the water’s icy cold, a beach walk with loved ones can still warm your heart. That salty breeze, soft sand, and laughter? Pure coastal therapy

Absolutely! The waves might be chilly, but the joy of a beach stroll never fades. Sometimes, the best memories are made with just sand beneath your feet and good company by your side.

Indeed. Cold waters may pause the swim, but not the serenity. A mindful walk on the shore still activates every sense, sight, sound, rhythm. Sometimes, the pause is the point.

Exactly! You don’t need to dive in to feel the magic of the sea.

A walk along the shore, wind in your face, sand under your feet, that’s already its own kind of happiness.

The sea doesn’t always have to be for swimming, sometimes it’s just the soundtrack to a good stroll.

Absolutely! Cold water or not, the beach always delivers

When the sea says "not today" with its icy waves, the shore still welcomes you warmly.

A peaceful beach walk > a cold swim any day!

Nothing bad about beach walking, that breeze still heating while you remain dry.
#gmfrens #freecompliments

Beautiful! I took a similar photo of yellow flowers.

This should be Dandelions. They are usually spotted on grassy lands. Beautifully captured.

Good morning Onealfa.

What a view to start threading. Good morning

The contrast between the delicate yellow blooms and the vast calm sea is just.

Such peaceful coastal vibes, nature always finds a way to bloom beautifully, even in the sand.

These tiny yellow flowers standing strong near the shore… pure poetry.

Beauty takes place in unusual places and life doesn't stop. Nice photo catch!

There’s something magical about wildflowers thriving near the sea.

Good morning Sir

Very beautiful and raw.

Excellent photography to notice and capture. Good morning Sir

They are beautiful flowers and the sea nearby makes them more special. Good morning Sir

They can grow in beautiful places. Just need to identify with our appropriative eyes.

A magical raw beauty. Excellent photography

There’s something magical about little yellow flowers by the sea…
So delicate, yet they thrive in the wind and salt.

Tiny bursts of sunshine holding their own by the sea.

Funny how something so delicate can thrive where the winds never stop.

A gentle pop of yellow in the wild grasses… like sunshine growing right out of the sand

Beautiful

tHe flowers are beautiful

USA?

This is very true.

That’s beautifully said, it’s amazing how everything around and within us vibrates in harmony or dissonance. Frequencies shape our experiences more than we realize. Off to read your post now!

It turns out, one strategy still holds strong, and the other is branching into ghostwriting."

Yeah, this month has to be a good one. All the best.

Yes! Gonna start farming some new airdrops today! 😅💪

That was brilliant,when I first heard about it I dug into it more.

Love to see Putin's face when he found out haha

That’s wild! Looks like they’re trying to contact Mars or maybe just get the clearest signal in the forest! Props to the passion, that’s serious dedication to the craft.

Must have cost a lot. Those amateurs have some money.

Gotta admire the dedication of radio amateurs they don’t play when it comes to signal strength

Also... imagine building that in snow! Total respect.

This looks like a boat mast, amazing construction. I bet he can reach quite far away, right?

That's not an antenna that's an engineering marvel disguised as a winter tree ornament. Absolutely wild what passion and persistence can build.

Expert they are specially in snow. Nice photo. Good morning

Just looking at the picture freezes me lol intelligent crafting

very tall, dedication at its peak to create such things.

Simply incredible I'd say. Ancient way of connecting to people of unknown lands.

It shivers me by the sight of it. Great skills

Indeed crazy . How many days of dedication to create it.

It's time to engage and rethreads to have enough Leo to subscribe to the premium.

Good morning Sir

I thought it was a ski lift at first. Radio amateurs really said: go big or go completely off-grid! This thing probably picks up signals from Moon.

Looks effective to have a transmission.

Wow, that's massive! That antenna looks like it could talk to aliens on a snowy mountaintop! Radio amateurs really take their passion to sky-high levels, literally!

Impressive engineering. Precision, persistence, and passion, hallmarks shared by both surgeons and radio amateurs. Signals, like synapses, need structure to bridge distance and meaning.

I thought it was part of a ski resort at first, turns out it’s just passion and precision by radio lovers! Gotta admire the dedication.

That’s not an antenna that’s a winter wonderland skyscraper for signals!

Honestly, hats off to the passion and precision, some hobbies really do reach for the sky, literally!

That’s absolutely insane and kinda beautiful in a frozen, sci-fi kind of way!

Amateur? More like radio architects!

Really huge

https://inleo.io/hive-166847/@drax/television-review-spectre-of-the-gun-star-trek-s3x01-1968-7md

Including the SEC dropping cases against Binance and more! 2/3 https://img.leopedia.io/DQme7uyth7oszLarwdc9vpmoR96izf4d3yUBJ7w4iJRkVNX/Trump%E2%80%99s%20SEC%20Retreats,%20TON%20Falters%20and%20Privacy%20Coins%20Surge%20(1).jpg

find out more on #InLeo built on the #Hive blockchain 3/3

https://inleo.io/@melbourneswest/trumps-sec-retreats-ton-falters-and-privacy-coins-surge-c2t

I remember 2 way radios were a novel way to communicate back in the day. Lots of hobbyiest use it sometimes 24/7.

That’s incredible dedication! Over 3,000 counties via radio and your friend did them all four times? Now that’s next-level DXing!

Meanwhile, this McDonald’s map adds such a quirky contrast. Fast food vs. fast signals.

That’s seriously impressive! Over 3,000 counties? That’s a whole radio tour of America.

Crazy number of US counties and amazing that you've got in touch using radio waves with them. This world has no limits!

That’s not dedication, that’s legendary status. Also, love how this McDonald's map accidentally reveals just how vast and under-commercialized parts of the U.S. still are.

Huge achievement 👏 takes years to do this.

Quite educating

Hive is still undervalued in many ways and using some to stack LEO? Smart move.

Looking at other tokens, I think it's a good time to do the buy

I'm adding Hive and targeting 10k HP this year.

Do what your gut says

We expecting huge profits

I said it on X, who fluffing cares about James Wynn