17Calculus - Limits of Vector Functions

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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.

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 $$\displaystyle{ \vec{r}(t) = \frac{\sin(t)}{t}\hat{i} + \frac{t+1}{3t+4}\hat{j} + \frac{\ln(t^2)}{t^3}\hat{k} }$$

$$\displaystyle{ \langle 0, 1/3, 0 \rangle }$$

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

$$\displaystyle{ \langle 0, 1/3, 0 \rangle }$$

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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 $$\displaystyle{ \lim_{t\to 0}{ \left( e^{-3t}\vhati + \frac{t^2}{\sin^2t}\vhatj + \cos 2t \vhatk \right)}}$$

$$\vhati + \vhatj + \vhatk$$

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

$$\vhati + \vhatj + \vhatk$$

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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 $$\displaystyle{ \lim_{t\to 0}{ \left[ e^t\vhat{i} + \frac{\sin t}{t}\vhat{j} \right] } }$$.

$$\hat{i} + \hat{j}$$

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

$$\hat{i} + \hat{j}$$

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

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

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

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

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

$$(1/2)\vhat{i} + \vhat{j} + (4/\pi)\vhat{k}$$

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

$$(1/2)\vhat{i} + \vhat{j} + (4/\pi)\vhat{k}$$

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