Limits of Vector Functions

Definition: If ${\vec {r}}(t)=(x(t),y(t),z(t))$ is a vector-valued function, then $\lim _{t\to a}{\vec {r}}(t)=\left(\lim _{t\to a}x(t),\lim _{t\to a}y(t),\lim _{t\to a}z(t)\right)$ provided that the limits of the components exist.

//Limits of vector-valued functions in $\mathbb {R} ^{n}$ are defined similarly as the limit of each component. //

Let's look at some examples of evaluating limits of vector-valued functions. Consider the vector-valued function ${\vec {r}}(t)=(t^{2}-1,t+1,e^{t})$ and suppose that we wanted to compute $\lim _{t\to 2}{\vec {r}}(t)$. To compute this limit, all we need to do is compute the limits of the components.

${\begin{aligned}\quad \lim _{t\to 2}{\vec {r}}(t)=\left(\lim _{t\to 2}t^{2}-1,\lim _{t\to 2}t+1,\lim _{t\to 2}e^{t}\right)=(3,3,e^{2})\end{aligned}}$

For another example, consider the vector-valued function ${\vec {r}}(t)=\left({\frac {e^{t}-1}{t}},{\frac {t-1}{t+1}},t^{2}+3\right)$ and suppose that we wanted to compute $\lim _{t\to 0}{\vec {r}}(t)$. To compute this limit, we will compute all of the limits of the components again, however, this time the limits are a little trickier to compute. Fortunately, we have already learned about various rules to evaluate limits.

${\begin{aligned}\quad \lim _{t\to 0}{\vec {r}}(t)=\left(\lim _{t\to 0}{\frac {e^{t}-1}{t}},\lim _{t\to 0}{\frac {t-1}{t+1}},\lim _{t\to 0}t^{2}+3\right)\end{aligned}}$

For $\lim _{t\to 0}{\frac {e^{t}-1}{t}}$ we will use L'Hospital's Rule, and so $\lim _{t\to 0}{\frac {e^{t}-1}{t}}{\overset {H}{=}}\lim _{t\to 0}{\frac {e^{t}}{1}}=1$.

For $\lim _{t\to 0}{\frac {t-1}{t+1}}$, we can use direct substitution and so $\lim _{t\to 0}{\frac {t-1}{t+1}}=-1$.

Now $\lim _{t\to 0}t^{2}+3$ is also easy to compute by direct substitution and so $\lim _{t\to 0}t^{2}+3=3$.

Thus we have that $\lim _{t\to 0}{\vec {r}}(t)=(1,-1,3)$.

The following theorem gives us a formal definition to say a vector-valued function ${\vec {r}}(t)$ has limit ${\vec {b}}$ at $t=a$, which is analogous to that of limits of real-valued functions.

Theorem 1: Let ${\vec {r}}(t)=(x(t),y(t),z(t))$ be a vector-valued function and let ${\vec {b}}=(b_{1},b_{2},b_{3})\in \mathbb {R} ^{3}$. Then $\lim _{t\to a}{\vec {r}}(t)={\vec {b}}$ if and only if $\forall \epsilon >0$ $\exists \delta >0$ such that if $0<\mid t-a\mid <\delta$ then $\|{\vec {r}}(t)-{\vec {b}}\|<\epsilon$.

• Proof: $\Rightarrow$ Suppose that $\lim _{t\to a}{\vec {r}}(t)={\vec {b}}$. Then we have that:

${\begin{aligned}\left(\lim _{t\to a}x(t),\lim _{t\to a}y(t),\lim _{t\to a}z(t)\right)=(b_{1},b_{2},b_{3})\end{aligned}}$

• Now recall that two vectors are equal if and only if their components are equal, and so the equation above implies that $\lim _{t\to a}x(t)=b_{1}$, $\lim _{t\to a}y(t)=b_{2}$, and $\lim _{t\to a}z(t)=b_{3}$. Now notice that these three limits are limits of real-valued functions.

• Since $\lim _{t\to a}x(t)=b_{1}$ then $\forall \epsilon >0$ $\exists \delta _{1}>0$ such that if $0<\mid t-a\mid <\delta _{1}$ then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid x(t) - b_1 \mid < \frac{\epsilon}{3}} .

• Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{t \to a} y(t) = b_2} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall \epsilon > 0} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists \delta_2 > 0} such that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < \mid t - a \mid < \delta_2} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid y(t) - b_2 \mid < \frac{\epsilon}{3}} .

• Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{t \to a} z(t) = b_3} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall \epsilon > 0} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists \delta_3 > 0} such that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < \mid t - a \mid < \delta_3} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid z(t) - b_3 \mid < \frac{\epsilon}{3}} .

• Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta = \mathrm{min} \{ \delta_1, \delta_2, \delta_3 \}} . Then if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < \mid t - a \mid < \delta} we have that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \quad \| \vec{r}(t) - \vec{b} \| = \| (x(t) - b_1, y(t) - b_2, z(t) - b_3)) \| = \sqrt{(x(t) - b_1)^2 + (y(t) - b_2)^2 + (z(t) - b_3)^2} \\ \quad \quad \leq \sqrt{(x(t) - b_1)^2} + \sqrt{(y(t) - b_2)^2} + \sqrt{(z(t) - b_3)^2} = \mid x(t) - b_1 \mid + \mid y(t) - b_2 \mid + \mid z(t) - b_3 \mid < \frac{\epsilon}{3} + \frac{\epsilon}{3} + \frac{\epsilon}{3} = \epsilon \end{align}}

• $\Leftarrow$ Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall \epsilon > 0} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists \delta > 0} such that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < \mid t - a \mid < \delta} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \| \vec{r}(t) - \vec{b} \| < \epsilon} . Therefore we have that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \| \vec{r}(t) - \vec{b} \| = \| (x(t) - b_1, y(t) - b_2, z(t) - b_3) \| < \epsilon} , which implies that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sqrt{(x(t) - b_1)^2 + (y(t) - b_2)^2 + (z(t) - b_3)^2} < \epsilon \\ (x(t) - b_1)^2 + (y(t) - b_2)^2 + (z(t) - b_3)^2 < \epsilon^2 \\ \end{align}}

• Now since all terms of the lefthand side of this equation are positive, we must have that for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < \mid t - a \mid < \delta} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x(t) - b_1)^2 < \epsilon^2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y(t) - b_2)^2 < \epsilon^2} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (z(t) - b_3)^2 < \epsilon^2} , and so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid x(t) - b_1 \mid < \epsilon} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid y(t) - b_2 \mid < \epsilon} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid z(t) - b_3 \mid < \epsilon} . Therefore by the definition of real-valued function limits we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{t \to a} x(t) = b_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{t \to a} y(t) = b_2} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{t \to a} z(t) = b_3} .

• Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{t \to a} \vec{r}(t) = \left ( \lim_{t \to a} x(t), \lim_{t \to a} y(t), \lim_{t \to a} z(t) \right ) = (b_1, b_2, b_3) = \vec{b}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}

Licensing

Content obtained and/or adapted from:

• Limits of Vector-Valued Functions, mathonline.wikidot.com under a CC BY-SA license