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27 June 2003

When Things Get One-Sided

by George E. Hrabovsky, President, MAST

Another Look at Our Problem Involving a Rational Function

Let's assume you have analyzed a data set and discovered that you have a pattern in the data described by the formula,

y = (| x |)/x .

We see that as we get closer to x = 0 something a bit unsettling happens.  We can see it if we plot this,

[Graphics:HTMLFiles/index_3.gif]

It seems as though we have two different values when x = 0, depending on which direction you are coming from.  This means that no function exists at this point.  No limit exists at this point, either.

One-Sided Limits

What we have just seen is an example of a special requirement for a limit to exist.  We used this intuitively up to now.  If we start from the negative direction of the independent variable of a function and discover a limit, that is called a limit to the left.  We can then write,

Underscript[lim, x  a^-] f(x) = L^-.

If we start from the positive direction and work towards the negative and discover a limit, then we have a limit to the right.  We write this,

Underscript[lim, x  a^+] f(x) = L^+.

These are called one-sided limits.  It turns out that,

Underscript[lim, x  a] f(x) = L

if and only if

Underscript[lim, x  a^-] f(x) = L^-.

and

Underscript[lim, x  a^+] f(x) = L^+.

such that

L^- = L^+.

The Math Challenge from Last Time

The challenge was to prove the triangle inequality,

| x + y | ≤ | x | + | y | .

By definition, we have

-| a | ≤ a≤ | a |,

so,

-| x | ≤ x≤ | x |,

and

-| y | ≤ y≤ | y | .

We can define inequality such that

a ≤ b ⟹ a - b ≤ 0,

then

a ≤ b ∧ c ≤ d ⟹ a + c ≤ b + d,

so we have,

-| x | - | y | ≤ x + y ≤ | x | + | y | .

This is the definition of an absolute value, | x + y |. So, we now have

| x + y | ≤ | x | + | y | .

Which is what we wanted to prove.

The second part was to show the relationships between the trigonometric ratios.  We use the definitions based on the unit circle from last time.

sin θ = y,

cos θ = x,

tan θ = y/x = (sin θ)/(cos θ),

cot θ = x/y = (cos θ)/(sin θ) = tan^(-1) θ

sec θ = 1/x = 1/(cos θ) .

csc θ = 1/y = 1/(sin θ) .

The Math Challenge

Can you prove the assertion that for a limit to exist its right and left limits must be the same?

Math Resources for Limits

Online:

http://www.ping.be/~ping1339/lim.htm

This is not a graphically interesting page, but it seems to be very good for the math content.

or, for a more advanced treatment,

http://www.gap-system.org/~john/analysis/Lectures/L13.html

Books:

Richard A. Silverman, 1969, Modern Calculus and Analytic Geometry, MacMillan Company, New York (Dover Publications has reprinted this book with corrections in 2002). This has a very nice chapter on limits.

Created by Mathematica  (June 27, 2003)