Limits, L'Hôpital's rule, and epsilon delta definitions | Chapter 7, Essence of calculus

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Recommends Mike H · 2018-03 · Mathematics Stack Exchange · on “Resources to make people understand the concepts of limits and continuity.”

As a high school student who plans to go on to study physics, I feel somewhat qualified to understand the point of view of your students. For me, understanding continuity is all about understanding motion. I feel like showing how all motion is continuous would help physics undergrads come to a better understanding. They should catch on pretty quick to see what it means, as the motion of an object can't be undefined . This could be demonstrated with a powerpoint presentation pretty easily, maybe just show a small clip of a ball being thrown through the air and relate it to continuity of a function in the plane. Limits might be a little more tricky, but if they have any aspirations on studying

Recommends Kyle · 2019-08 · Mathematics Stack Exchange · on “how to apply "$ \varepsilon$- $\delta$" definition of limit”

I’m glad googling images made it clear! As you start to get into more and more advanced math, it’s always a good idea to hang on to some intuitive understanding of what’s going on. If you need help understanding the intuition behind limits and their definition, I’d suggest this video here: [this video] (as a disclaimer: I actually haven’t watched this video myself, but I’ve seen enough of his other videos to trust him on this one.)

Recommends Joe · 2020-11 · Mathematics Stack Exchange · on “Why is there a condition on L'Hopital's Rule?”

A while back, 3blue1brown posted a video [this video] that seeks to explain why L'Hôpital's rule works intuitively. The video focuses on $0/0$—perhaps others can explain why L'Hôpital's rule works in the case of $\infty/\infty$. Also, I think it is helpful to shift one's mindset. By default, *don't* expect something to work until it has been proven otherwise.

Cites user366312 · 2018-06 · Mathematics Stack Exchange · on “Why is a derivative defined using limits?”

From our childhood, we learn mathematics along with a gradual progression where topics are sequentially related to each other. The discussion of calculus, almost, always starts with the concept of limits [this video] . This is where most beginners start to gasp for air. The concept of limit is a disconnected concept which superficially connects algebra and calculus. Why is a derivative defined using the concept of limit?

Cites PrincessEev · 2018-11 · Mathematics Stack Exchange · on “What is a plain English explanation of "$\epsilon$-$\delta$ criterion" and how to use it for proving continuity?”

A nice way of envisioning it, in my opinion, is by what my real analysis professor called " $\epsilon-\delta$ bands." Imagine choosing two points $x,y$ on the horizontal axis of the graph of a function $f$ . Drawing vertical lines, you will find $f(x), f(y)$ where the lines cross the graph - nothing new there. We define the distance between $x$ and $y$ by $\delta$ and the distance between $f(x)$ ad $f(y)$ by $\epsilon$ . Mathematically: $$|x - y| = \delta \;\;\;\;\; \text{and} \;\;\;\;\; |f(x)-f(y)| = \epsilon$$ My professor would have said then that we have bands of these widths. An example of what I'm talking about: [LINK] Imagine some sort of function $f$ in your mind that you know to be

Cites AltercatingCurrent · 2022-04 · Mathematics Stack Exchange · on “Will the L'Hopital's Rule work here, when it agrees with the derivation of the rule as I know it?”

Say I have to find the limit for: $$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}$$ Such that $$g(a)=0≠f(a)$$ Multiplying the numerator and denominator by $g(x)$ , I get $$\lim_{x\rightarrow a}\frac{f(x)g(x)}{g(x)^2}$$ and I get the indeterminate form $\frac 00$ . So can I use the L'Hopital's Rule to find the limit here? From 3B1B's video about L'Hopital's Rule at time stamp 14:30 [this video] , the rule works as the value of the numerator graph and denominator graph at given value which $x$ approaches is $0$ , so I can just divide their derivatives. So will the L'Hopital's Rule work here, considering I can just derive the rule from such arrangement of the graphs?

Critical Not Really · 2019-07 · Mathematics Stack Exchange · on “L'Hospital's rule only works if the functions are "straight"?”

I think I get how L'Hospital's rule works, at an intuitive level based on the ratios of the derivatives being equivalent to the ratios of the functions and therefore the limit at a point where they would otherwise evaluate to 0/0, so I am happy there. However while learning about it I see something odd in certain explanations, like here: [LINK] and here [this video] If the derivatives are known, then this further criteria or observation of functions "looking like a line" or "being straight" when you "zoom in" doesn't really mean anything here does it? Either you have the derivatives where f(x)/g(x) = 0/0 or you don't. And if you do, L'Hopital's rule applies. Since we are not using infinitesimals