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# 10.2 Limits of Functions

Our definition of tangent to a curve is going to be based on the idea of limit. The word limit was used in mathematics long before the definition we will give was thought of. One finds statements like  The limit of a regular polygon when the number of sides becomes infinite, is a circle." Early definitions of limit often involved the ideas of time or motion. Our definition will be purely mathematical.

10.5   Definition (Interior points and approachable points.) Let be a subset of . A point is an interior point of if there is some positive number such that the interval is a subset of . A point is an approachable point from if there is some positive number such that either or . (Without loss of generality we could replace  " in this definition by for some .)

Note that interior points of must belong to . Approachable points of need not belong to . Any interior point of is approachable from .

10.6   Example. If is the open interval then every point of is an interior point of . The points that are approachable from are the points in the closed interval .

If is the closed interval then the points that are approachable from are exactly the points in , and the interior points of are the points in the open interval .

10.7   Definition (Limit of a function.) Let be a real valued function such that . Let and let . We say
 (10.8)

if
1)
is approachable from , and
2)
For every sequence in

Note that the value of (if it exists) has no influence on the meaning of . Also the  " in (10.8) is a dummy variable, and can be replaced by any other symbol that has no assigned meaning.

10.9   Example. For all we have

Also

and

by lemma 9.34. Also

by theorem 9.37.

10.10   Example. is not defined. Let . Then is a sequence in , and and . We know there is no number such that .

10.11   Example. Let be the spike function

Then , since if is a generic sequence in , then is the constant sequence .

10.12   Example. The limit

does not exist. If , then the domain of consists of the single point , and is not approachable from . If we did not have condition 1) in our definition, we would have

which would not be a good thing. (If there are no sequences in , then

is true, no matter what statement about is.)

In this course we will not care much about functions like .

10.13   Example. I will show that
 (10.14)

for all .

Case 1: Suppose . Let be a generic sequence in such that . Then

Now, since , we have

so by the squeezing rule which is equivalent to

This proves (10.14) when .

Case 2: Suppose . The domain of the square root function is , and is approachable from this set.

Let be a sequence in such that . To show that , I'll use the definition of limit. Let . Then , so by the definition of convergence, there is an such that for all we have . Then for all we have and hence .

Many of our rules for limits of sequences have immediate corollaries as rules for limits of functions. For example, suppose are real valued functions with and . Suppose and . Let be a generic sequence in such that . Then is a sequence in and , so

Also is a sequence in and so

By the sum and product rules for sequences, for any

and

and thus we've proved that

and

Moreover if (so that ), and if for all (so that for all ), it follows from the quotient rule for sequences that

so that

Actually all of the results just claimed are not quite true as stated. For we have

and

but

The correct theorem is:

10.15   Theorem (Sum, product, quotient rules for limits.) Let
be real valued functions with and , and let . Suppose , and both exist. Then if is approachable from we have

Proof: Most of the theorem follows from the remarks made above. We will assume the remaining parts.

10.16   Theorem (Inequality rule for limits of functions.) Let and be real functions with and . Suppose that
i
and both exist.
ii
is approachable from .
iii
There is a positive number such that
for all in .
Then .

Proof: Let be a sequence in such that . Then is a sequence in that converges to , so by the definition of limit of a function,

Similiarly

Also for all , so it follows from the inequality rule for limits of sequences that , i.e. .

10.17   Theorem (Squeezing rule for limits of functions.) Let , and be real functions with , , and . Suppose that
i
and both exist and are equal.
ii
is approachable from .
iii
There is a positive number such that for all in
.
Then .

Proof: The proof is almost identical to the proof of theorem 10.16.

Next: 10.3 Definition of the Up: 10. Definition of the Previous: 10.1 Velocity and Tangents   Index
Ray Mayer 2007-09-07