Limits and Continuity
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The concept of a 'limit' applies whenever we are interested in the way a quantity behaves close to, but not exactly at, a point of interest. The point that we are approaching may be perfectly normal, or it may be abnormal in some way. For example, if we stretch a wire to breaking point, we may not be able to measure the length exactly when it breaks, but we have plenty of oportunity to make careful measurements for loads just below the breaking point.
In many cases the limiting behaviour of a function can be easily read off from its graph. Some examples are illustrated in animations by Doug Arnold at Penn State (you can choose either an animated gif or a java version). An important point made in these examples is that the existence and value of the limit do not depend on the value of the function at the limiting point.
We also use the language of limits to describe the eventual or 'limiting' behaviour of a quantity as some variable like time or distance becomes extremely large. For example, the population of a certain species introduced into a new environment might be expected to rise at first and then to level off and gradually approach some finite limiting value (or in other circumstances it might rise up to an excessive level and then have a catastrophic falloff - perhaps repeating the pattern over and over again).
In terms of the graph, if there is in fact a stable limiting population, then the graph of population vs time will have a horizontal asymptote.
A similar concept applies when we are trying to approximate something which we can't calculate exactly. We may have a sequence of successively better approximations and want to define the 'exact' answer as a 'limit' of these. Examples of this include successive decimal approximations of a number (as in the calculation of 1/3 by long division), or Archimedes' successive approximations to the area of a circle.
This tutorial from
a college teacher in California starts with Archimedes, and includes also
more illustrations of all of the above types of behaviour.
of e as a limit is illustrated graphically by a java applet from the
IES group in Japan.
They also have applets demonstrating a couple of others, including the limit of sinx /x as x->0.
And Doug Arnold also has a demo of sinx /x
Although an intuitive/graphical concept of limit is adequate for most
purposes, there are some more subtle questions for which we need to make
it clearer exactly what we mean. The formal precise definition of limits
is often not studied in a calculus class but left instead to a course in
'Analysis' (at Langara that is Math2373).
On-line discussions of the formal definition of limits for sequences
and for functions
are included in the 'Interactive Real Analysis' course notes at Seton Hall
University, and also in the Analysis
WebNotes by John Orr at the University of Nebraska.
Just as for limits, an intuitive sense of what continuity means will often suffice. A more precise formal definition and JAVA applet illustrating the concept can be found at the Continuity section of Interactive Real Analysis, or you can look at Continuous functions, in the Analysis WebNotes site. The epsilon-delta definition is given, and an interactive demonstration can be used to explore the consequences of using different values of d for a given e.
An important fact about continuous functions is the Intermediate Value Theorem. This basically says that if f is continuous throughout the interval [a,b] then every value between f(a) and f(b) must be taken at least once in between x=a and x=b, so that if f(a)<c<f(b), then there must be an x in (a,b) satisfying the equation f(x)=c. (Or in other words, a graph with no breaks can't get from y=f(a) to y=f(b) without passing through every y-value in between). This seems like a simple idea, but can be very useful for proving the existence of solutions to equations even if we can't actually solve them, and it can also guide us towards a sequence of better and better approximate solutions as demonstrated in this lesson on the "Bisection Method".
You might also check our 'raw list' (of links provided without comment) to see if there are any more examples there that we haven't yet included here.
If you have come across any good web-based illustrations of these and
please do let us know and we will add them here.
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