Calculus Calculus Math Forum

March 21st, 2019, 04:12 PM   #11
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Quote:
 Originally Posted by NineDivines The most embarrassing thing about this post is not on the OP but the glorified “Math Team” who don’t give a sh*t. Who can spot what’s wrong with the OP’s definition of derivative? How on Earth that slipped past by this so called “team” I don’t know. Either give up your gloves for the sake of your reputation or be active to correct people.
My thinking was that:

$\displaystyle \frac{\Delta{y}}{\Delta{x}} = \frac{f(x+h) - f(x)}{h}$

So if I multiply both sides with lim:
$\displaystyle \frac{\lim_{h \to \infty}\Delta{y}}{\lim_{h \to \infty} \Delta{x}} = \lim_{h \to \infty} \frac{f(x+h) - f(x)}{h}$

And this is why I assumed that:
$\displaystyle \lim_{h \to \infty} \Delta{x} = dx \Leftrightarrow \lim_{x_0-x \to \infty} \Delta{x} = dx$

The I was told that $\displaystyle \lim_{x_0-x \to \infty} \Delta{x} = 0$ which is true. They also told me that $\displaystyle \frac{d}{dx}$ is just an operator (symbol).

I got the idea now not entirely but I will try to understand better next time I'll re-read calculus, having in mind what I was told here.

Last edited by babaliaris; March 21st, 2019 at 04:16 PM. March 21st, 2019, 05:51 PM   #12
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Quote:
 Originally Posted by babaliaris So if I multiply both sides with lim: $\displaystyle \frac{\lim_{h \to \infty}\Delta{y}}{\lim_{h \to \infty} \Delta{x}} = \lim_{h \to \infty} \frac{f(x+h) - f(x)}{h}$ Your limit is as h goes to 0, not as h goes to infinity.

You are not "multiplying both sides with lim." lim is an operator. It acts on both sides of the equation but it is NOT a number!

Yes, it is true that $\displaystyle \lim_{h \to 0} \dfrac{\Delta y}{ \Delta x} \to \dfrac{dy}{dx}$ but bear in mind that this is not an equality. The limit of $\displaystyle \dfrac{ \Delta y}{ \Delta x}$ as h goes to 0 is $\displaystyle \dfrac{dy}{dx}$ is what is true.

It sounds like a minor point of language but I assure you it is critical in your thinking. There is no value h such that $\displaystyle \dfrac{ \Delta y}{ \Delta x} = \dfrac{dy}{dx}$

-Dan Tags Δx, limit Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post zylo Calculus 13 May 31st, 2017 01:53 PM yo79 Math Events 1 February 16th, 2014 08:28 AM yo79 Math Events 3 November 23rd, 2013 09:42 AM veronicak5678 Real Analysis 4 August 22nd, 2011 11:07 AM conjecture Calculus 1 July 24th, 2008 02:14 PM

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