Key to "Give an example of a vector-valued function that..."

Elma Wilson

Answered question

2021-11-25

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

t = 3

Geraldine Flores

Beginner2021-11-26Added 21 answers

Step 1
Given

r (t) = ⟨ 2 t + 1, \frac{t^{2} - 9}{t - 3}, t^{2} ⟩

Here the domain of r excludes the value

t = 3

The limit does not exist at

t \to 3

lim_{t \to 3} r (t) = lim_{t \to 3} ⟨ 2 t + 1, \frac{t^{2} - 9}{t - 3}, t^{2} ⟩

lim_{t \to 3} r (t) = ⟨ lim_{t \to 3} (2 t + 1), lim_{t \to 3} \frac{t^{2} - 9}{t - 3}, lim_{t \to 3} t^{2} ⟩

= ⟨ lim_{t \to 3} (2 t + 1), lim_{t \to 3} \frac{(t - 3) (t + 3)}{t - 3}, lim_{t \to 3} t^{2} ⟩

lim_{t \to 3} r (t) = ⟨ 2 (3) + 1, (3 + 3), {(3)}^{2} ⟩

lim_{t \to 3} r (t) = ⟨ 7, 6, 9 ⟩

Step 2
The middle expression simplifies as

\frac{t^{2} - 9}{t - 3} = \frac{(t - 3) (t + 3)}{t - 3} = 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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