Give an example of a vector-valued function that is defined

Elma Wilson

Elma Wilson

Answered question

2021-11-25

Give an example of a vector-valued function that is defined but not continuous at t=3

Answer & Explanation

Geraldine Flores

Geraldine Flores

Beginner2021-11-26Added 21 answers

Step 1
Given r(t)=2t+1, t29t3, t2
Here the domain of r excludes the value t=3
The limit does not exist at t3
limt3r(t)=limt32t+1, t29t3, t2
limt3r(t)=limt3(2t+1), limt3t29t3, limt3t2
=limt3(2t+1), limt3(t3)(t+3)t3, limt3t2
limt3r(t)=2(3)+1, (3+3), (3)2
limt3r(t)=7, 6, 9
Step 2
The middle expression simplifies as t29t3=(t3)(t+3)t3=t+3
Then the limit is taken.
However, the value t=3 is still excluded from the domain, so r is not continuous at t=3.
There is a deleted point in the curve when t=3

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