Calculus of Vector Valued Finctions

Limits and Continuity Derivatives Integrals Arclength Exercises

Calculus of Vector-Valued Functions

Mathematics 54 - Elementary Analysis 2

Institute of Mathematics

University of the Philippines-Diliman

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Limits and Continuity Derivatives Integrals Arclength Exercises

Definition Examples

Limits and Continuity of Vector Functions

Definition.

Given R (t) = x (t) , y (t) , z (t) .

1

We define the limit of R as t approaches a by

limR (t) = limx (t) , limy (t) , limz (t) ,

t→a

t→a

t→a

t→a

provided that lim x (t), lim y (t), and lim z (t) exist.

t→a

2

t→a

t→a

The function R (t) is continuous at t = a if

R (a) exists;

lim R (t) exists;

t→a

R (a) = lim R (t).

t→a

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Limits and Continuity Derivatives Integrals Arclength Exercises

Definition Examples

Limits of Vector Functions

Example

Evaluate the following limits:

1

limR (t) where R (t) = t + 1,

t→2

2

lim R (t) where R (t) =

t→1−

1

t 2 − 4 sin (2t − 4)

,

.

t −2

t −2

|t − 1| sin(πt) tan(πt)

,

,

.

t − 1 t2 − 1

t −1

We have

limR (t)

t→2

t2 − 4

sin (2t − 4)

, lim

t→2

t→2 t − 2 t→2

t −2

2 cos (2t − 4)

= 3, lim (t + 2) , lim

(L’Hopital’s Rule)

t→2

t→2

1

= 〈3, 4, 2〉 .

= lim (t + 1) , lim

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Limits and Continuity Derivatives Integrals Arclength Exercises

Definition Examples

Limits of Vector Functions

2

Note that for t → 1− , t < 1. We have

lim R (t)

−

t→1

=

=

lim

−

t→1

sin(πt)

tan(πt)

|t − 1|

, lim 2

, lim

−

−

t − 1 t→1 t − 1 t→1 t − 1

lim

−

t→1

− (t − 1)

sin(πt)

tan(πt)

, lim 2

, lim

−

−

t − 1 t→1 t − 1 t→1 t − 1

= −1, lim

−

t→1

π cos(πt)

π sec2 (πt)

, lim

−

t→1

2t

1

(L’Hopital’s Rule)

π

= −1, − , π .

2

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Limits and Continuity Derivatives Integrals Arclength Exercises

Definition Examples

Continuity of Vector Functions

Example

Determine whether the function



 sin(t) , t − 1, et , t = 0...