Expert: Paul Klarreich - 9/5/2007

Question
I'm 15 and taking Calculus this year. We did a little bit with limits and derivatives and such last year, but 1. the teacher wasn't very good, so 2. I taught myself most of the stuff... Anyway, this year we have a crazy strict teacher, who wants us to write a paragraph on "how to determine if a particular limit exists without using GRAPHING technology." I'm not asking you to write my paragraph for me, but could you help me on the stuff I don't have? It would've been so much easier if she had asked only about fraction functions, but unfortunately she didn't, and I'm not sure how to generalize other kinds. This is what I have so far:

To determine if the limit lim (x->n)(f(x))  exists without graphing f(x), first evaluate f(n). One of three things will happen:
1.   the expression is a real number over zero;
2.   the expression is zero over zero.
3.   the expression evaluates to a real number;
In 1, the limit does not exist; in 2, factoring is required. Once one has factored the expression f(x), which in math problems will usually end up canceling out the lower term, evaluate the simplified expression for n. In 3, however…

And that's it.

P.S. It's due tomorrow...

Answer
Questioner:   Brianna
Category:  Calculus
Private:  No
 
Subject:  Determining Limit Existence Without Graphing
Question:  I'm 15 and taking Calculus this year. We did a little bit with limits and derivatives and such last year, but 1. the teacher wasn't very good, so 2. I taught myself most of the stuff... Anyway, this year we have a crazy strict teacher, who wants us to write a paragraph on "how to determine if a particular limit exists without using GRAPHING technology." I'm not asking you to write my paragraph for me, but could you help me on the stuff I don't have? It would've been so much easier if she had asked only about fraction functions,

>> The proper term is RATIONAL FUNCTIONS -- a polynomial on top and bottom.


but unfortunately she didn't, and I'm not sure how to generalize other kinds.

This is what I have so far:

To determine if the limit lim (x->n)(f(x))  exists without graphing f(x), first evaluate f(n). One of three things will happen:
1. the expression is a real number over zero;
2. the expression is zero over zero.
3. the expression evaluates to a real number;
In 1, the limit does not exist; in 2, factoring is required. Once one has factored the expression f(x), which in math problems will usually end up canceling out the lower term, evaluate the simplified expression for n. In 3, however…

And that's it.

P.S. It's due tomorrow...
..................................
Hi, Brianna,

WELL, if it's due tomorrow, I'd better hurry.

Now the definition of:

lim f(x) = L
x->a

is that f(x) is close to L  WHENEVER  x is close to a.  (Be sure to use that word WHENEVER in your paragraph.  Teachers love that.)

Unfortunately, finding f(a) does not help an awful lot, because:

1. f(a) might exist, but the limit might not exist.

2. f(a) might not exist, but the limit might.

I think your paragraph starts with that sentence above, and could say something like this:

======== START OF PARAGRAPH ==========
lim(x->a) f(x) = L  means that f(x) is close to L  WHENEVER  x is close to a.  So to determine whether the limit exists, study  | f(x) - L |.  If making  | x - a | very small is always enough to make  | f(x) - L | small, then the limit exists and is equal to L.
======= END ==========

In the formal sense (which your teacher probably does not want now), you write:

Given a (small) number called epsilon, (I will write e) there exists a (small) number delta (I write d) such that | f(x) - L | < e  WHENEVER  | x - a | < d.  (there's that word again -- I love it.)

I hope this helps.