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[[Image:limits_of_functions_h.png]] | [[Image:limits_of_functions_h.png]] | ||
| − | We can see from the graphs that <math> | + | We can see from the graphs that <math>f</math> is "continuous" at <math>0</math> and that <math>g, h</math> are "discontinuous" at 0. But, what exactly do we mean? Intuitively, <math>f</math> seems to be continuous at <math> 0</math> because <math>f(x)</math> is close to <math>f(0)</math> whenever <math>x</math> is close to <math> 0</math>. On the other hand, <math>g</math> appears to be ''discontinuous'' at <math> 0</math> because there are points <math>x</math> which are close to <math>0</math> but such that <math>g(x)</math> is far away from <math>g(0)</math>. |
| − | Let us make these observations more precise. In each of the figures above, we have marked off a band of width <math>1/2</math> with two dashed red lines. The red dots correspond to the point <math> | + | Let us make these observations more precise. In each of the figures above, we have marked off a band of width <math>1/2</math> with two dashed red lines. The red dots correspond to the point <math>x=0</math>. |
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Revision as of 23:34, 10 May 2014
Contents
Tutorial Template
by: Michael Yeh, proud Member of the Math Squad.
keyword: tutorial, limit, function, sequence
INTRODUCTION Provided here is a brief introduction to the concept of "limit," which features prominently in calculus. We first discuss the limit of a function at a point; to help motivate the definition, we first consider continuous functions. Unless otherwise mentioned, all functions here will have domain and range $ \mathbb{R} $, the real numbers.
Continuous functions
Let's consider the the following three functions along with their graphs (in blue). The red markings will be explained later.
- $ \displaystyle f(x)=x^3 $
- $ g(x)=\begin{cases}-x^2-\frac{1}{2} &\text{if}~x<0\\ x^2+\frac{1}{2} &\text{if}~x\geq 0\end{cases} $
- $ h(x)=\begin{cases} \sin\left(\frac{1}{x}\right) &\text{if}~x\neq 0\\ 0 &\text{if}~x=0\end{cases} $
We can see from the graphs that $ f $ is "continuous" at $ 0 $ and that $ g, h $ are "discontinuous" at 0. But, what exactly do we mean? Intuitively, $ f $ seems to be continuous at $ 0 $ because $ f(x) $ is close to $ f(0) $ whenever $ x $ is close to $ 0 $. On the other hand, $ g $ appears to be discontinuous at $ 0 $ because there are points $ x $ which are close to $ 0 $ but such that $ g(x) $ is far away from $ g(0) $.
Let us make these observations more precise. In each of the figures above, we have marked off a band of width $ 1/2 $ with two dashed red lines. The red dots correspond to the point $ x=0 $.
TOPIC 3
Lorem Ipsum [1] is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy text ever since the 1500s, when an unknown printer took a galley of type and scrambled it to make a type specimen book. It has survived not only five centuries, but also the leap into electronic typesetting, remaining essentially unchanged. It was popularised in the 1960s with the release of Letraset sheets containing Lorem Ipsum passages, and more recently with desktop publishing software like Aldus PageMaker including versions of Lorem Ipsum.
TOPIC 2
Lorem Ipsum [1] is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy text ever since the 1500s, when an unknown printer took a galley of type and scrambled it to make a type specimen book. It has survived not only five centuries, but also the leap into electronic typesetting, remaining essentially unchanged. It was popularised in the 1960s with the release of Letraset sheets containing Lorem Ipsum passages, and more recently with desktop publishing software like Aldus PageMaker including versions of Lorem Ipsum.
REFERENCES
[1] "Loream Ipsum" <http://www.lipsum.com/>.
Questions and comments
If you have any questions, comments, etc. please, please please post them below:
- Comment / question 1
- Comment / question 2
The Spring 2013 Math Squad 2013 was supported by an anonymous gift to Project Rhea. If you enjoyed reading these tutorials, please help Rhea "help students learn" with a donation to this project. Your contribution is greatly appreciated.
