Limits of functions - Rhea

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. Words such as "all," "every," "each," "some," and "there are" are quite important here; read carefully!

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) $. The same observation applies to $ h $.

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 $, e.g. for $ f $, the red dot is the point $ (0,f(0))=(0,0) $.

TOPIC 3

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TOPIC 2

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REFERENCES

[1] "Loream Ipsum" <http://www.lipsum.com/>.

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