Lecture notes on Limit and continuity of vector- valued functions

Lecture notes on Limit and continuity of vector- valued functions

by Shilpa Patra

Department of Mathematics Narajole Raj College

West bengal, India

Vector valued Function

Definition( Vector-valued Function)

A vector-valued function is a function of the form

r(t) =f(t)i+g(t)jor r(t) =f(t)i+g(t)j+h(t)k,

where the component functions f , g , and h are real-valued functions of the parameter t .

Vector-valued functions are also written in the form

r(t) =hf(t),g(t)ior r(t) =hf(t),g(t),h(t)i.

In both cases, the first form of the function defines a two-dimensional vector-valued function; the second form describes a three-dimensional vector-valued function.

Example

r(t) = (t²−1)i+ (2t−3)j,0≤t≤3.

r(t) = 4 costi+ 4 sintj+tk,0≤t≤4π.

Graphing a Vector valued Function

Sketch 1

The graph of the following vector functionsr(t) =ht,1iis

Sketch 2

LIMIT OF A VECTOR-VALUED FUNCTION

Definition

A vector-valued functionrapproaches the limit Las t approaches a, written

t→alimr(t) =L, provided

t→alimkr(t)−Lk= 0.

means that given any >0 , there exists aδ >0 such that for allt 6=a , if|t−a|< δ , we havekr(t)−Lk ≤.

Theorem:

Theorem

Let f , g , and h be functions of t . Then the limit of the vector-valued function

r(t) =f(t)i+g(t)j as t approaches a is given by

t→alimr(t) = [ lim

t→af(t)]i+ [ lim

t→ag(t)j, provided the limits lim_t→af(t) and lim_t→ag(t) exist.

Similarly, the limit of the vector-valued function r(t) =f(t)i+g(t)j+h(t)k as t approaches a is given by

t→alimr(t) = [ lim

t→af(t)]i+ [ lim

t→ag(t)]j+] lim

t→ah(t)]i,

provided the limits limt→af(t) and limt→ag(t) and limt→ah(t) exist.

Useful Rules:

Theorem

Assuming that lim_t→aA(t), lim_t→aB(t), and lim_t→af(t) exists, then 1. lim_t→a(A±B)(t) = lim_t→aA(t)±lim_t→aB(t),

2. lim_t→a[fA](t) = lim_t→af(t) lim_t→aA(t), 3. lim_t→a(A·B)(t) = lim_t→aA(t)·lim_t→aB(t), 4.limt→a(A×B)(t) = limt→aA(t)×limt→aB(t).

Evaluating Limit of a Vector-Valued Function

Example1

calculate lim_t→3r(t) for

r(t) = (t²−3t+ 4)i+ (4t+ 3)j.

Solution

t→3limr(t) = lim

t→3

h

(t²−3t+ 4)i+ (4t+ 3)ji

= h

t→3lim(t²−3t+ 4)i i+h

t→3lim(4t+ 3)i j

= 4i+ 15j.

Example 2

calculate limt→3r(t) for r(t) =2t−4

t+ 1

i+ t t²+ 1

j+ (4t−3)k.

Solution

t→3limr(t) = lim

t→3

h (2t−4

t+ 1 )i+ ( t

t²+ 1)j+ (4t−3)ki

= [ lim

t→3

2t−4

t+ 1 )]i+ [ lim

t→3

t

t²+ 1]j+ [ lim

t→3(4t−3)]k

= 1

2i+ 3 10j+ 9k.

Continuity of a Vector-Valued Function

Definition

Let f , g , and h be functions of t . Then, the vector-valued function r(t) =f(t)i+g(t)j+h(t)k

is continuous at point t=a if the following three conditions hold:

r(a) exists, lim_t→ar(t) exists, lim_t→ar(t) =r(a).

Theorem

A vector-valued functionris continuous at a if and only if each of its component functions is continuous at a.

Example 3

The vector valued function

r(t) = 2 costi+sint

t j+t²k.

is discontinuous at t=0.

Theorem

IfA(t) is continuous att0and lim_s→s0g(s) =t0. Then

s→slim0

(A(g(s))) =A( lim

s→s0

g(s)) =A(t0).

Testing Continuity of a Vector-Valued Function

Example 4 Given

r(t) =t²i+e^tj+ 1 t+ 3k.

Find lim_t→2r(t). Isrcontinuous at t= 2 ? Solution: The limit is

t→2limr(t) = lim

t→2

ht²i+e^tj+ 1 t+ 3ki

= [ lim

t→2t²]i+ [ lim

t→2e^t]j+ [ lim

t→2

1 t+ 3]k

= 4i+e²j+1

5k=r(2).

Since all three conditions of continuity are met, soris continuous at t=

2. In this example, the limit ofrast → −3 does not exist since the limit fails to exist for the expression _t+3¹ . This curve is not continuous when t= -3. It is continuous everywhere else.