On this page we explain how to evaluate limits of vector functions.
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Start with this video on limits of vector functions. This video contains great explanations and examples.
video by Dr Chris Tisdell 

Let's try some practice problems.
Practice
Unless otherwise instructed, evaluate these limits giving your answers in exact form.
\(\displaystyle{\lim_{t \to 0}{ \left\langle \frac{e^t1}{t}, \frac{\sqrt{1+t}1}{t}, \frac{3}{1+t} \right\rangle }}\)
Problem Statement
Evaluate the limit \(\displaystyle{\lim_{t \to 0}{ \left\langle \frac{e^t1}{t}, \frac{\sqrt{1+t}1}{t}, \frac{3}{1+t} \right\rangle }}\)
Solution
video by PatrickJMT 

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Find \(\displaystyle{ \lim_{t\to4}{ \vec{r}(t) } }\) for \(\displaystyle{ \vec{r}(t) = (4t)\hat{i} + \sqrt{12+t}\hat{j}  [ \cos(\pi t/8) ]\hat{k} }\)
Problem Statement
Find \(\displaystyle{ \lim_{t\to4}{ \vec{r}(t) } }\) for \(\displaystyle{ \vec{r}(t) = (4t)\hat{i} + \sqrt{12+t}\hat{j}  [ \cos(\pi t/8) ]\hat{k} }\)
Solution
video by PatrickJMT 

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Find \(\displaystyle{ \lim_{t\to\infty}{ \vec{r}(t) } }\) for \(\displaystyle{ \vec{r}(t) = \frac{\sin(t)}{t}\hat{i} + \frac{t+1}{3t+4}\hat{j} + \frac{\ln(t^2)}{t^3}\hat{k} }\)
Problem Statement 

Find \(\displaystyle{ \lim_{t\to\infty}{ \vec{r}(t) } }\) for \(\displaystyle{ \vec{r}(t) = \frac{\sin(t)}{t}\hat{i} + \frac{t+1}{3t+4}\hat{j} + \frac{\ln(t^2)}{t^3}\hat{k} }\)
Final Answer 

\(\displaystyle{ \langle 0, 1/3, 0 \rangle }\)
Problem Statement
Find \(\displaystyle{ \lim_{t\to\infty}{ \vec{r}(t) } }\) for \(\displaystyle{ \vec{r}(t) = \frac{\sin(t)}{t}\hat{i} + \frac{t+1}{3t+4}\hat{j} + \frac{\ln(t^2)}{t^3}\hat{k} }\)
Solution
video by PatrickJMT 

Final Answer
\(\displaystyle{ \langle 0, 1/3, 0 \rangle }\)
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\(\displaystyle{ \lim_{t\to 0}{ \left( e^{3t}\vhati + \frac{t^2}{\sin^2t}\vhatj + \cos 2t \vhatk \right)}}\)
Problem Statement 

Evaluate the limit \(\displaystyle{ \lim_{t\to 0}{ \left( e^{3t}\vhati + \frac{t^2}{\sin^2t}\vhatj + \cos 2t \vhatk \right)}}\)
Final Answer 

\( \vhati + \vhatj + \vhatk \)
Problem Statement
Evaluate the limit \(\displaystyle{ \lim_{t\to 0}{ \left( e^{3t}\vhati + \frac{t^2}{\sin^2t}\vhatj + \cos 2t \vhatk \right)}}\)
Solution
video by Krista King Math 

Final Answer
\( \vhati + \vhatj + \vhatk \)
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\(\displaystyle{ \lim_{t\to 0}{ \left[ e^t\vhat{i} + \frac{\sin t}{t}\vhat{j} \right] } }\)
Problem Statement 

Evaluate the limit \(\displaystyle{ \lim_{t\to 0}{ \left[ e^t\vhat{i} + \frac{\sin t}{t}\vhat{j} \right] } }\).
Final Answer 

\( \hat{i} + \hat{j} \)
Problem Statement
Evaluate the limit \(\displaystyle{ \lim_{t\to 0}{ \left[ e^t\vhat{i} + \frac{\sin t}{t}\vhat{j} \right] } }\).
Solution
video by MIP4U 

Final Answer
\( \hat{i} + \hat{j} \)
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\(\displaystyle{ \lim_{t\to\infty}{ \left[ \frac{2}{t}\vhat{i} + \frac{t^3}{2t^38}\vhat{j} + e^{t}\vhat{k} \right] } }\)
Problem Statement 

Evaluate the limit \(\displaystyle{ \lim_{t\to\infty}{ \left[ \frac{2}{t}\vhat{i} + \frac{t^3}{2t^38}\vhat{j} + e^{t}\vhat{k} \right] } }\).
Final Answer 

\( (1/2)\vhat{j} \)
Problem Statement
Evaluate the limit \(\displaystyle{ \lim_{t\to\infty}{ \left[ \frac{2}{t}\vhat{i} + \frac{t^3}{2t^38}\vhat{j} + e^{t}\vhat{k} \right] } }\).
Solution
video by MIP4U 

Final Answer
\( (1/2)\vhat{j} \)
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\(\displaystyle{ \lim_{t\to\pi/4}{ \left[ \sin^2(t)\vhat{i} + \tan(t)\vhat{j} + \frac{1}{t}\vhat{k} \right] } }\)
Problem Statement 

Evaluate the limit \(\displaystyle{ \lim_{t\to\pi/4}{ \left[ \sin^2(t)\vhat{i} + \tan(t)\vhat{j} + \frac{1}{t}\vhat{k} \right] } }\).
Final Answer 

\( (1/2)\vhat{i} + \vhat{j} + (4/\pi)\vhat{k} \)
Problem Statement
Evaluate the limit \(\displaystyle{ \lim_{t\to\pi/4}{ \left[ \sin^2(t)\vhat{i} + \tan(t)\vhat{j} + \frac{1}{t}\vhat{k} \right] } }\).
Solution
video by MIP4U 

Final Answer
\( (1/2)\vhat{i} + \vhat{j} + (4/\pi)\vhat{k} \)
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Practice Instructions
Unless otherwise instructed, evaluate these limits giving your answers in exact form.