Difference between revisions of "Limits of functions" - Rhea

Revision as of 22:26, 10 May 2014

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 close to 0 whose image under g is far away from g(0), and similarly for h.

Let us make these observations more precise. In each of the figures above, we have marked off a band of width 0.5 with two dashed red lines. The red dots correspond to the point $ x=0 $.

TOPIC 3

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TOPIC 2

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REFERENCES

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