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17Calculus - Limits of Vector Functions

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

On this page we explain how to evaluate limits of vector functions.

Start with this video on limits of vector functions. This video contains great explanations and examples.

Dr Chris Tisdell - limits of vector functions [44mins-37secs]

video by Dr Chris Tisdell

Let's try some practice problems.

How to Ace the Rest of Calculus: The Streetwise Guide, Including MultiVariable Calculus

Practice

Unless otherwise instructed, evaluate these limits giving your answers in exact form.

\(\displaystyle{\lim_{t \to 0}{ \left\langle \frac{e^t-1}{t}, \frac{\sqrt{1+t}-1}{t}, \frac{3}{1+t} \right\rangle }}\)

Problem Statement

Evaluate the limit \(\displaystyle{\lim_{t \to 0}{ \left\langle \frac{e^t-1}{t}, \frac{\sqrt{1+t}-1}{t}, \frac{3}{1+t} \right\rangle }}\)

Solution

PatrickJMT - 698 video solution

video by PatrickJMT

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

Problem Statement

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

Solution

PatrickJMT - 701 video 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

PatrickJMT - 702 video 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

Krista King Math - 2029 video 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

MIP4U - 2031 video solution

video by MIP4U

Final Answer

\( \hat{i} + \hat{j} \)

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\(\displaystyle{ \lim_{t\to\infty}{ \left[ \frac{2}{t}\vhat{i} + \frac{t^3}{2t^3-8}\vhat{j} + e^{-t}\vhat{k} \right] } }\)

Problem Statement

Evaluate the limit \(\displaystyle{ \lim_{t\to\infty}{ \left[ \frac{2}{t}\vhat{i} + \frac{t^3}{2t^3-8}\vhat{j} + e^{-t}\vhat{k} \right] } }\).

Final Answer

\( (1/2)\vhat{j} \)

Problem Statement

Evaluate the limit \(\displaystyle{ \lim_{t\to\infty}{ \left[ \frac{2}{t}\vhat{i} + \frac{t^3}{2t^3-8}\vhat{j} + e^{-t}\vhat{k} \right] } }\).

Solution

MIP4U - 2032 video 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

MIP4U - 2033 video 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.

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