MthT 430 Notes Chapter 5a Limits

MthT 430 Notes Chapter 5a Limits

Notation

The expression

lim
x ® a

f(x) = L

is read

·: The limit of f at x = a is L.

·: The limit as x approaches a of f(x) is L.

·: The limit of f(x) is L as x approaches a.

·: f(x) approaches L as x approaches a.

·: The function f approaches the limit L near a (Note: no mention of x).

·: (Briefer - p. 99) f approaches L near a.

Meaning

The meaning of the phrase is

Provisional Definition. (p. 90) The function f approaches the limit L near a, if we can make f(x) as close as we like to L by requiring that x be sufficiently close to (but ¹ ) a.

·: (Somewhat Informal) The function f approaches the limit L near a, if f(x) - L is small whenever x - a is small enough (but x ¹ a).

·: (Different Words - Somewhat Informal) The function f approaches the limit L near a, if f(x) = L + small whenever x = a+ small enough (but x ¹ a).

·: (Informal) The function f approaches the limit L near a, if f(x) is close to L whenever x is close enough to (but ¹ ) a.

·: (Explanation of Provisional) You tell me how close you want f(x) to be to L and I will tell you how close x needs to be to a to force f(x) to be as close to L as you requested.

·: (Explanation of Different Words - Somewhat Informal) f(x) = L + small means that size of f(x) - L is small in the sense that, f(x)- L is as small as we like (whether .1, .00001, 10^-100, ¼), by imposing that |x - a| is small enough (but ¹ 0). How small is small enough for x - a depends on how small we require f(x) - L to be.

·: (More Explanation of Provisional JL) Given a positive size [number] e, there is a positive size [number] d such that if the size of x - a is less than d (but not 0, then the size of f(x) - L is less than e. Here the size of a number is its absolute value.

Definition of Limit

Definition. (p. 96) The function f approaches the limit L near a means: For every e > 0, there is some d > 0 such that, for all x, if 0 < |x -a| < d, then |f(x) - L| < e.

Different Words. (p. 96) The function f approaches the limit L near a means: For every desired degree of closeness e > 0, there is a degree of closeness d > 0 such that, for all x ¹ a, if x -a is within dof a, then f(x) is within eof L.
The phrase a is within e of b means: |a- b| < e.

Change of Notation. The function f approaches the limit L near a means: For every § > 0, there is some © > 0 such that, for all ª, if 0 < |ª-a| < ©, then |f(ª) - L| < §.
Fundamental Properties of Limits

Theorem 1. The limit is unique. If f approaches L near a, and f approaches M near a, then L = M.
Informal Proof: For x near enough to a, f(x) is very close to both L and M. By the triangle inequality,

|L - M|

= |(L - f(x)) + (f(x) - M)|

£ |L - f(x)| + |f(x) - M|

= small+ small

= small.

Thus for x - a small enough, |L - M| is as small as desired. Conclude L = M.

Fact. A number Y = 0 iff for very e > 0, |Y| < e.
Proof: (Text, p. 98.)

Theorem 2. If lim_{x ® a} f(x) = L and lim_{x ® a} g(x) = M, then

lim
x ® a
(f + g)(x)

= L + M,

lim
x ® a
(f ·g)(x)

= L ·M.

If M ¹ 0, then

lim
x ® a
æ
è 1
g
ö
ø (x) = 1
M
.

Proof. See Spivak, Problems 1.20 ff.

Discussion before the proof: Let's do the result for products. We can make (how? - by requiring x - a to be small enough (and ¹ 0) f(x) = L +small_f and g(x) = M + small_g. Then for x = a +small enough, x ¹ a,

f(x) ·g(x)

= (L + small_f)·(M +small_g)

= L ·M + small_f ·M + L ·small_g + small_f ·small_g

= L ·M + Remainder.

Now it is evident that Remainder can be made as small as we like by requiring |x - a| sufficiently small (but ¹ 0).

The Proof: Given e > 0, we have

| f(x) ·g(x) - L ·M|

= |small_f ·M + L ·small_g + small_f ·small_g|,

where small_f = f(x) - L, small_g = g(x) - M. Now choose d > 0 so that whenever 0 < |x - a| < d,

|small_f| = |f(x) - L|

< e,

|small_g|=|g(x) - M|

< e.

Then whenever 0 < |x - a| < d,

| f(x) ·g(x) - L ·M|

= |small_f ·M + L ·small_g + small_f ·small_g|,

£ |e·M| + |e·L| + e².

(*)

Now if we also assume that e < 1, we have that

(*) £

e·( |M| + |L| + 1),

and it is evident that |f(x) - L| can be made as small as desired. There are a couple of ways:

·: Choose the d that works for [^(e)] = [(e)/(( |M| + |L| + 1))] > 0.

·: Use a modified equivalent definition of limit: The function f approaches the limit L near a means: There is an e₀ > 0 and a K > 0 such that : For every e, e₀ > e > 0, there is a d > 0 such that, for all x, if 0 < |x -a| < d, then |f(x) - L| < K ·e.

Notes

·: Given e > 0, the d such that 0 < |x - a| < d assures |f(x) - L| < e usually depends on e, as well as depending on the point a and function f and all of its properties. Finding an explicit expression for the optimal d is not required nor necessarily interesting unless doing numerical error estimates.

·: In the product and quotient example, the d = d_e was chosen with the additional requirement that e < 1.

·: Pay attention to the domain of the function. See the technical detail on p. 102.

·: Observe the definitions of one sided limits - also called limits from above [below] and limits from the left [right] .

Thinking About Limits

Definition. (Actual, p. 96)

lim
x ® a
f(x) = L

means: For every e > 0, there is some d > 0 such that, for all x, if 0 < |x -a| < d, then |f(x) - L| < e.

Definition - Working JL II.

lim
x ® a
f(x) = L

means:

·

For all x = a + smallenufneq0, x is in domainf.

·

f(x) = L + assmallasdesired, for x = a + smallenufneq0.
Translation:

·: assmallasdesired means, given e > 0, then |assmallasdesired| < e is the desired result.
·: smallenufneq0 means, find a d > 0 such that 0 < |smallenufneq0| < d is the sufficient condition.
·: smallenuf means, find a d > 0 such that |smallenufneq0| < d is the sufficient condition.

Definition - Working JL II'.

lim
x ® a
f(x) = L

means:

·

For |x - a| smallenufneq0,x is in domainf.

·

f(x) - L is assmallasdesired,for x - a is smallenufneq0.

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On 02 Oct 2007, 10:05.