Limits and Continuity of Vector Functions
The idea is now to extend the notion of a limit of a function and continuity of a function to vector functions by capitalizing on the linearity properties of vector functions.
Definition (Limit of Vector-Valued Functions) Suppose the components function
![limits and continuity of vector functions _gr_1.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_1.gif){kind=small}
,
![limits and continuity of vector functions _gr_2.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_2.gif){kind=small}
, and
![limits and continuity of vector functions _gr_3.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_3.gif){kind=small}
of the vector-valued function
![limits and continuity of vector functions _gr_4.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_4.gif){kind=small}
all have finite limits as
![limits and continuity of vector functions _gr_5.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_5.gif){kind=small}
, where
![limits and continuity of vector functions _gr_6.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_6.gif){kind=small}
is any real number or
![limits and continuity of vector functions _gr_7.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_7.gif){kind=small}
. Then the limit of
![limits and continuity of vector functions _gr_8.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_8.gif){kind=small}
as
![limits and continuity of vector functions _gr_9.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_9.gif){kind=small}
is the vector
![limits and continuity of vector functions _gr_10.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_10.gif){kind=small}
Notice that if the limit of a vector function exists then it is a vector, (and not a scalar function).
Proposition (Limits of Vector-Valued Functions) If the vector-valued functions
![limits and continuity of vector functions _gr_11.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_11.gif){kind=small}
and
![limits and continuity of vector functions _gr_12.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_12.gif){kind=small}
are functions of a real variable
![limits and continuity of vector functions _gr_13.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_13.gif){kind=small}
and
![limits and continuity of vector functions _gr_14.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_14.gif){kind=small}
is a real-valued function such that all three functions have finite limits as
![limits and continuity of vector functions _gr_15.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_15.gif){kind=small}
, then
![limits and continuity of vector functions _gr_16.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_16.gif){kind=small}
![limits and continuity of vector functions _gr_17.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_17.gif){kind=small}
![limits and continuity of vector functions _gr_18.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_18.gif){kind=small}
![limits and continuity of vector functions _gr_19.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_19.gif){kind=small}
![limits and continuity of vector functions _gr_20.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_20.gif){kind=small}
These limit formulas are also valid when
![limits and continuity of vector functions _gr_21.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_21.gif){kind=small}
provided all limits are finite.
Example (Limits of Vector-Valued Functions) Given
![limits and continuity of vector functions _gr_22.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_22.gif){kind=small}
and
![limits and continuity of vector functions _gr_23.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_23.gif){kind=small}
find
![limits and continuity of vector functions _gr_24.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_24.gif){kind=small}
Solution. We have,
![limits and continuity of vector functions _gr_25.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_25.gif){kind=small}
![limits and continuity of vector functions _gr_26.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_26.gif){kind=small}
Example (Limits of Vector-Valued Functions) Given
![limits and continuity of vector functions _gr_27.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_27.gif){kind=small}
and
![limits and continuity of vector functions _gr_28.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_28.gif){kind=small}
find
![limits and continuity of vector functions _gr_29.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_29.gif){kind=small}
Solution. We have,
![limits and continuity of vector functions _gr_30.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_30.gif){kind=small}
![limits and continuity of vector functions _gr_31.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_31.gif)
![limits and continuity of vector functions _gr_32.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_32.gif){kind=small}
![limits and continuity of vector functions _gr_33.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_33.gif){kind=small}
Definition (Continuity of Vector-Valued Functions) A vector-valued function is continuous at
![limits and continuity of vector functions _gr_34.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_34.gif){kind=small}
means
![limits and continuity of vector functions _gr_35.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_35.gif){kind=small}
is in the domain of
![limits and continuity of vector functions _gr_36.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_36.gif){kind=small}
and
![limits and continuity of vector functions _gr_37.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_37.gif){kind=small}
. Further, a vector function is continuous on an interval
![limits and continuity of vector functions _gr_38.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_38.gif){kind=small}
if it is continuous at every point in the interval.
Example (Continuity of Vector-Valued Functions) Determine where the vector-valued function
![limits and continuity of vector functions _gr_39.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_39.gif){kind=small}
is continuous.
Solution. The function
![limits and continuity of vector functions _gr_40.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_40.gif){kind=small}
is continuous for all real numbers in its domain which is
![limits and continuity of vector functions _gr_41.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_41.gif){kind=small}
![limits and continuity of vector functions _gr_42.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_42.gif){kind=small}
Example (Continuity of Vector-Valued Functions) Determine where the vector-valued function
![limits and continuity of vector functions _gr_43.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_43.gif){kind=small}
where
![limits and continuity of vector functions _gr_44.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_44.gif){kind=small}
is continuous.
Solution. The function
![limits and continuity of vector functions _gr_45.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_45.gif){kind=small}
is continuous for all real numbers in its domain which is
![limits and continuity of vector functions _gr_46.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_46.gif){kind=small}
because
![limits and continuity of vector functions _gr_47.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_47.gif)
and
![limits and continuity of vector functions _gr_48.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_48.gif){kind=small}
![limits and continuity of vector functions _gr_49.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_49.gif)
![limits and continuity of vector functions _gr_50.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_50.gif)
![limits and continuity of vector functions _gr_51.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_51.gif)
for all real numbers such that
![limits and continuity of vector functions _gr_52.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_52.gif){kind=small}
![limits and continuity of vector functions _gr_53.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_53.gif){kind=small}
![limits and continuity of vector functions _gr_54.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_54.gif){kind=small}
Recommended Links
![limits and continuity of vector functions _gr_55.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_55.gif){kind=small}
vector functions
![limits and continuity of vector functions _gr_56.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_56.gif){kind=small}
operations with vector functions
![limits and continuity of vector functions _gr_57.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_57.gif){kind=small}
limits and continuity of vector functions
![limits and continuity of vector functions _gr_58.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_58.gif){kind=small}
graphs of vector functions
![limits and continuity of vector functions _gr_59.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_59.gif){kind=small}
vector differentiation
![limits and continuity of vector functions _gr_60.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_60.gif){kind=small}
tangent vectors
![limits and continuity of vector functions _gr_61.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_61.gif){kind=small}
smooth curves
![limits and continuity of vector functions _gr_62.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_62.gif){kind=small}
derivative rules
![limits and continuity of vector functions _gr_63.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_63.gif){kind=small}
vector integration
![limits and continuity of vector functions _gr_64.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_64.gif){kind=small}
motion of an object
![limits and continuity of vector functions _gr_65.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_65.gif){kind=small}
unit tangent and unit normal vectors
![limits and continuity of vector functions _gr_66.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_66.gif){kind=small}
arc length function
![limits and continuity of vector functions _gr_67.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_67.gif){kind=small}
curvature
![limits and continuity of vector functions _gr_68.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_68.gif){kind=small}
Recommended Math Books
![limits and continuity of vector functions _gr_69.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_69.gif){kind=small}
Thomas' Calculus, Early Transcendentals, Media Upgrade (11th Edition)
![limits and continuity of vector functions _gr_70.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_70.gif){kind=small}
Thomas' Calculus, Media Upgrade (11th Edition)
![limits and continuity of vector functions _gr_71.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_71.gif){kind=small}
Thomas' Calculus Early Transcendentals; Student's Solutions Manual; Part One
![limits and continuity of vector functions _gr_72.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_72.gif){kind=small}
Calculus (With Analytic Geometry)(8th edition)
![limits and continuity of vector functions _gr_73.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_73.gif){kind=small}
Calculus (Stewart's Calculus Series)
![limits and continuity of vector functions _gr_74.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_74.gif){kind=small}
Applied Calculus
![limits and continuity of vector functions _gr_75.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_75.gif){kind=small}
Calculus Textbooks
![limits and continuity of vector functions _gr_76.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_76.gif){kind=small}
Elementary Calculus
![limits and continuity of vector functions _gr_77.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_77.gif){kind=small}
Advanced Calculus
![limits and continuity of vector functions _gr_78.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_78.gif){kind=small}
Supplementary Resources
![limits and continuity of vector functions _gr_79.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_79.gif){kind=small}
Recommended Math Gifts
![limits and continuity of vector functions _gr_80.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_80.gif){kind=small}
Math Happy
![limits and continuity of vector functions _gr_81.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_81.gif){kind=small}
Calculus Happy
![limits and continuity of vector functions _gr_82.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_82.gif){kind=small}
Homework Happy
![limits and continuity of vector functions _gr_83.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_83.gif){kind=small}
Limits Happy
![limits and continuity of vector functions _gr_84.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_84.gif){kind=small}
I Love Math
![limits and continuity of vector functions _gr_85.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_85.gif){kind=small}
I Love Calculus
![limits and continuity of vector functions _gr_86.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_86.gif){kind=small}
I Love Homework
![limits and continuity of vector functions _gr_87.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_87.gif){kind=small}
I Love Multivariate Calculus
![limits and continuity of vector functions _gr_88.gif]](pages/limits-and-continuity-of-vector-functions/Images/limits-and-continuity-of-vector-functions_gr_88.gif){kind=small}
I Love Limits
Limits And Continuity Of Vector Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/limits-and-continuity-of-vector-functions.html
