What is the best to solve first order differential equations?

Asked by Raju P P 13/10/2017 Last Modified 07/10/2025

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Hi friends. I thought of sharing something wonderful i found on web so that it can help all of us & augment our understanding of the topic - 'Limit' . I have shared the complete pdf in my profile gallery or the original web link at the end of this post. An Intuitive Introduction To Limits Limits, the Foundations Of Calculus, seem so artificial and weasely: “Let x approach 0, but not get there, yet we’ll act like it’s there… ” Ugh. Here’s how I learned to enjoy them: What is a limit? Our best prediction of a point we didn’t observe. How do we make a prediction? Zoom into the neighboring points. If our prediction is always in between neighboring points, no matter how much we zoom, that’s our estimate. Why do we need limits? Math has “black hole” scenarios (dividing by zero, going to infinity), and limits give us a reasonable estimate. How do we know we’re right? We don’t. Our prediction, the limit, isn’t required to match reality. But for most natural phenomena, it sure seems to. Limits let us ask “What if?”. If we can directly observe a function at a value (like x=0, or x growing infinitely), we don’t need a prediction. The limit wonders, “If you can see everything except a single value, what do you think is there?”. When our prediction is consistent and improves the closer we look, we feel confident in it. And if the function behaves smoothly, like most real world functions do, the limit is where the missing point must be. Key Analogy: Predicting A Soccer Ball (associated pics in original post) Pretend you’re watching a soccer game. Unfortunately, the connection is choppy: So we missed what happened at 4:00. Even so, what’s your prediction for the ball’s position? Easy. Just grab the neighboring instants (3:59 and 4:01) and predict the ball to be somewhere in between. And… it works! Real world objects don’t teleport? they move through intermediate positions along their path from A to B. Our prediction is “At 4:00, the ball was between its position at 3:59 and 4:01?. Not bad. With a slow motion camera, we might even say “At 4:00, the ball was between its positions at 3:59.999 and 4:00.001?. Limits are a strategy for making confident predictions. Limits aren’t the only tool for checking the answers to impossible questions; infinitesimals work too. The key is understanding what we’re trying to predict, then learning the rules of making predictions. Happy math. (Original author - Mr. kalid ) Original post link: http://betterexplained.com/articles/an-intuitive-introduction-to-limits/ My profile link: https://www.urbanpro.com/delhi/pankaj-k/2531974 You may find more interesting stuff and information that can be of some help to you. I will be adding more pdfs in gallery soon . Sharing is caring.

If one wants the pdf in my galley on 'limits' can give a better understanding of the same material.

Pankaj

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What is the best to solve first order differential equations?

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