Engineering Math

11.1 Review Of Limits

Recall the concept of limit of a function of many variables. When f : D

→ ℝ^q one can only consider in a meaningful way limits at limit points of the set D

Definition 11.1.1 Let A denote a nonempty subset of ℝ^p. A point x is said to be a limit point of the set A if for every r > 0,B

contains infinitely many points of A.

Example 11.1.2 Let S denote the set

. Which points are limit points?

This set does not have any because any two of these points are at least as far apart as 1. Therefore, if x is any point of ℝ³,B

contains at most one point.

Example 11.1.3 Let U be an open set in ℝ³. Which points of U are limit points of U?

They all are. From the definition of U being open, if x ∈ U, There exists B

⊆ U for some r > 0. Now consider the line segment x + tre₁ where t ∈

. This describes infinitely many points and they are all in B

because

= tr < r. Therefore, every point of U is a limit point of U.

The case where U is open will be the one of most interest, but many other sets have limit points.

Definition 11.1.4 Let f : D

⊆ ℝ^p → ℝ^q where q,p ≥ 1be a function and let x be a limit point of D

. Then

yli\tomx f (y) = L

if and only if the following condition holds. For all ε > 0 there exists δ > 0 such that if

0 < |y- x| < δ and y \in D (f)

then,

|L - f (y)| < ε.

The condition that x must be a limit point of D

if you are to take a limit at x is what makes the limit well defined.

Proposition 11.1.5 Let f : D

⊆ ℝ^p → ℝ^q where q,p ≥ 1be a function and let x be a limit point of D

. Then if lim_y→xf

exists, it must be unique.

Proof: Suppose lim_y→xf

= L₁ and lim_y→xf

= L₂. Then for ε > 0 given, let δ_i > 0 correspond to L_i in the definition of the limit and let δ = min

. Since x is a limit point, there exists y ∈ B

∩ D

. Therefore,

|L - L | \leq |L - f (y)|+ |f (y)- L | < ε+ ε = 2ε. 1 2 1 2

Since ε > 0 is arbitrary, this shows L₁ = L₂. ■

The following theorem summarized many important interactions involving continuity. Most of this theorem has been proved in Theorem 8.5.5 on Page 419.

Theorem 11.1.6 Suppose x is a limit point of D

and

lim f (y) = L, lim g (y ) = K y\tox y\tox

where K and L are vectors in ℝ^p for p ≥ 1. Then if a, b ∈ ℝ,

lim af (y)+ bg(y) = aL + bK, (11.1) y\tox

(11.1)

lim f \cdotg(y) = L \cdotK (11.2) y\tox

(11.2)

Also, if h is a continuous function defined near L, then

lim h\circ f (y) = h (L ). (11.3) y\tox

(11.3)

For a vector valued function

f (y) = (f1(y),\cdot\cdot\cdot,fq (y))T ,

lim_y→xf

= L =

^T if and only if

lim fk(y) = Lk (11.4) y\tox

(11.4)

for each k = 1,

,p.

In the case where f and g have values in ℝ³

lim f (y)\times g(y) = L \times K. (11.5) y\tox

(11.5)

Also recall Theorem 8.5.6 on Page 428.

Theorem 11.1.7 For f : D

→ ℝ^q and x ∈ D

such that x is a limit point of D

, it follows f is continuous at x if and only if lim_y→xf

= f