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Limits and Continuity.

Definition 2.5   Let $\overrightarrow{F(t)} =f_1(t)\overrightarrow{i} + f_2(t)\overrightarrow{j} +f_3(t)\overrightarrow{k} $ be a vector function, defined on the interval I, with values in the 3-dimensional space, and let $\overrightarrow{L} =l_1\overrightarrow{i} + l_2\overrightarrow{j} +l_3\overrightarrow{k} $ be a vector. We say that the vector $\overrightarrow{F(t)} $ approaches the vector $\overrightarrow{L} $ when t approaches t0, if:

\begin{displaymath}\forall \varepsilon > 0, \exists \delta > 0 \vert
\begin{vma...
...htarrow{F(t)} - \overrightarrow{L}\end{vmatrix} < \varepsilon.
\end{displaymath}

We denote:

\begin{displaymath}\underset{t \rightarrow t_0}{\lim}\overrightarrow{F(t)} = \overrightarrow{L} .
\end{displaymath}

Proposition 2.6  

\begin{displaymath}\underset{t \rightarrow t_0}{\lim}\overrightarrow{F(t)} = \ov...
... \\
\underset{t \rightarrow t_0}{\lim} f_3(t)=l_3
\end{cases}\end{displaymath}

Example 2.7   Let $\overrightarrow{F(t)} =(t-1)\overrightarrow{i} + t^2\overrightarrow{j} +\ln t \overrightarrow{k} $. Then:

\begin{displaymath}\underset{t \rightarrow 1}{\lim}\overrightarrow{F(t)} = 0 \; ...
...rrightarrow{j} + 0 \;\overrightarrow{k} = \overrightarrow{j} .
\end{displaymath}

Definition 2.8   The vector function $\overrightarrow{F(t)} $ is continuous at t0 if $\underset{t \rightarrow 1}{\lim}\overrightarrow{F(t)} = \overrightarrow{F(t_0)} $.

Proposition 2.9       

1.
The vector function $\overrightarrow{F(t)} $ is continuous at t0 if, and only if, each component Fi is continuous at t0.
2.
The vector function $\overrightarrow{F(t)} $ is continuous on the open interval I if it is continuous at every point of I.

Example 2.10       

1.
The function $\overrightarrow{F(t)} =(t-1)\overrightarrow{i} + t^2\overrightarrow{j} +\ln t \overrightarrow{k} $ is continuous on $(0,+\infty)$.
2.
The function $\overrightarrow{G(t)} =t \overrightarrow{i} + \sin t \overrightarrow{j} +\tan t \overrightarrow{k} $ is discontinuous at every point $\frac {pi}{2}+k \pi$ (with integer k), and continuous at every other point.

As in Calculus I, we can define one-sided continuity and continuity on a closed interval. Exercise: Have a look at the tutorial for Calculus I, and write down the corresponding definitions here.


next up previous contents
Next: Derivatives. Up: Curves in the plane Previous: Vector-valued functions.
Noah Dana-Picard
2001-05-30