Limit of a Vector-Valued Function

Topics: Vector-Valued Function - Calculus

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(definition)

Let $r(t):\mathbb{R}\to {\mathbb{R}}^n$ be a vector-valued function given by:

$$r(t)=⟨r_1(t),r_2(t),\dots ,r_n(t)⟩$$

We define the limit of $r(t)$ as $t$ approaches $a$ as:

$$\underset{t\to a}{\lim }r(t):=⟨\underset{t\to a}{\lim }{r}_1(t),\underset{t\to a}{\lim }{r}_2(t),\dots ,\underset{t\to a}{\lim }{r}_n(t)⟩$$

…provided all the indicated limits exist. If any one of them fails to exist, then ${\lim }_{t\to a}r(t)$ does not exist.