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I was working on some limit homework and everything was going fine until I reached this problem:

$$\lim_{\Delta x \to 0} \frac{2(x + \Delta x) - 2x}{\Delta x}.$$

I am understanding limits but the triangle (delta?) before the x is throwing me off. I have never seen it used this way before and I have no idea what it means in this context. Could anyone help me out? Thanks.

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So, $$\lim_{h \to 0} \frac{2(x + h) - 2x}{h}$$ would cause no trouble? Then simply read $h = \Delta x$ :) – t.b. May 24 '11 at 20:22
Yes, it is the capital of the Greek letter "delta." In this context, it is just part of a variable name. Every place you see $\Delta x$ you could replace it with $h$ or $\epsilon$ or otherwise. Mathematicians use $\Delta$ before a variable name to represent a "difference," usually a small one, in that variable. – Thomas Andrews May 24 '11 at 20:25
With the same meaning we can use $\delta x$ in $\lim_{\delta x\to 0}\dfrac{2(x+\delta x)-2x}{\delta x}$ or another letter instead of $h$, e.g. $\lim_{t\to 0}\dfrac{2(x+t)-2x}{t}$. The letter for the variable $h$ is irrelevant provided that it is different from the variable $x$. – Américo Tavares May 24 '11 at 20:40

2 Answers 2

up vote 11 down vote accepted

For the sake of having an answer: $\Delta x$ is just the name of a variable whose meaning is supposed to be "a small change in $x$." It is not, as I guess one might think, some kind of strange function of $x$ or anything like that.

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The Derivative of a function $f(x)$ denoted by $f'(x)$ is defined as the limit $$f'(x) = \lim_{\Delta{x} \to 0} \frac{f(x + \Delta{x}) - f(x)}{\Delta{x}}$$

So you limit is the derivative of the function $f(x)=2x$. Geometrically, derivative of a function can be seen in this picture: enter image description here

Good picture which was given at Wikipedia link: enter image description here

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@Theo: Geometrical interpretation of the derivative, Google Images – user9413 May 25 '11 at 7:58

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