1.4, Limit laws and continuity

Basic Limit Laws

The concept of the limit was originally developed to analyze behavior near a point where a function is really undefined.  In order to do this, we need some basic ideas about how limits work in routine situations.

For example, if we look at the limit, as x approaches 2, of the function

f(x) = 3x/(x^2+4) 

we should not be surprised to discover that 3x must approach 6 and x^2+4 must approach 8 and so the function approaches 6/8 = 3/4.  We do not need a calculator to argue this.

There are, then, some "obvious" limit laws.  They are described in the OpenStax textbook in section 2.3. (See the box there with 7 different identified laws!) The limit of a sum is the sum of the limits; the limit of a difference is the difference of the limits, etc.  Do NOT memorize these limit law, but merely recognize that the limit acts as you would expect!

We might summarize some of these laws by saying that, except for cases where the denominator is zero, the limit of a polynomial or rational function is "what we would expect" -- just plug the value in!  So the limit of

f(x) = 3x/(x^2+4) 

as x approaches 2 is just f(2) = 3/4.

Computing limits algebraically

If a function is not continuous at a particular point because the graph has a hole there, we can use algebra to remove the gap in the graph and then evaluate the limit by substitution.
For example the following are graphs of two functions.

The first graph is the the graph of f(t)=(t^2+2t-3)/(t^2-1).

The second graph is the graph of g(t) = (t+3)/(t+1).

What is the difference? The first graph has a removable singularity at t=1. Since limit computations only involve questions near a t-value, these two functions have the same limit as t approaches one.
So if we are asked to compute the limit at 1 of the first function, we use algebra to turn it into the second function (near t equal 1 but not at t=1.)  Here are the algebra computations:

The Squeeze Theorem

The "squeeze theorem" says that if a function g(x) is "squeezed" between two functions h(x) and f(x), so that f(x) < g(x) < h(x), and yet, at the place where x=a, the limit of the bottom function f(x) and the top function h(x) are the same, then the function g(x) must have the same limit.

(Download OpenStax materials for free at 

http://cnx.org/contents/8b89d172-2927-466f-8661-01abc7ccdba4@2.37.)

We will find numerous uses for this....

A trig limit

The squeeze theorem is useful finding a very important limit in trigonometry.  Note, from the geometry of the circle, that if the angle theta is close to zero then theta is between the sine of theta and the tangent of theta.

Continuity

A function f(x) is continuous at x=a if the limit of f(x) as x approaches a is equal to f(a).  This means that the limit is what we might expect, that the limit can be found by simply substituting the value a into the function.

Our algebra strategy, above, was to replace a function like  f(t)=(t^2+2t-3)/(t^2-1) by a function g(t) = (t+3)/(t+1) which is continuous at the desired value a. (Here a=1.) Once we have a continuous function, computing limits is easy!

The area of a circle

One early use of limits was in finding the area of a circle.  a nice short explanation of that can be found here.