## 17Calculus - Limits of Vector Functions

##### 17Calculus

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

Log in to rate this practice problem and to see it's current rating.

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

Log in to rate this practice problem and to see it's current rating.

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

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

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

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

Log in to rate this practice problem and to see it's current rating.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.