Problem 1

Question

In Exercises $1-4,$ find the given limits. $$\lim _{t \rightarrow \pi}\left[\left(\sin \frac{t}{2}\right) \mathbf{i}+\left(\cos \frac{2}{3} t\right) \mathbf{j}+\left(\tan \frac{5}{4} t\right) \mathbf{k}\right]$$

Step-by-Step Solution

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Answer

The limit is $ \left[ 1 \mathbf{i} - \frac{1}{2} \mathbf{j} + 1 \mathbf{k} \right] $.

1Step 1: Identify Components of the Vector

The given vector function is $ \left[\left(\sin \frac{t}{2}\right) \mathbf{i} + \left(\cos \frac{2}{3} t\right) \mathbf{j} + \left(\tan \frac{5}{4} t\right) \mathbf{k}\right] $. It consists of three components to take the limit of: $ \sin \frac{t}{2} $, $ \cos \frac{2}{3} t $, and $ \tan \frac{5}{4} t $. We will find the limit of each component separately as $ t \to \pi $.

2Step 2: Calculate the Limit of the First Component

Find $ \lim_{t \to \pi} \sin \frac{t}{2} $. When $ t = \pi $, $ \frac{t}{2} = \frac{\pi}{2} $, and $ \sin \frac{\pi}{2} = 1 $. Thus, the limit is $ 1 \mathbf{i} $.

3Step 3: Compute the Limit of the Second Component

Find $ \lim_{t \to \pi} \cos \frac{2}{3} t $. When $ t = \pi $, $ \frac{2}{3} t = \frac{2}{3} \pi $, and $ \cos \frac{2}{3} \pi = -\frac{1}{2} $. Thus, the limit is $-\frac{1}{2} \mathbf{j} $.

4Step 4: Determine the Limit of the Third Component

Find $ \lim_{t \to \pi} \tan \frac{5}{4} t $. When $ t = \pi $, $ \frac{5}{4} t = \frac{5}{4} \pi $, and $ \tan \frac{5}{4} \pi = \tan \left(\pi + \frac{\pi}{4}\right) = \tan \frac{\pi}{4} = 1 $. Thus, the limit is $1 \mathbf{k} $.

5Step 5: Combine Limits of Each Component

Combine the evaluated limits of the three components. The overall limit of the vector function as $ t \to \pi $ is $ 1 \mathbf{i} - \frac{1}{2} \mathbf{j} + 1 \mathbf{k} $. Thus, the limit is: $ \left[ 1 \mathbf{i} - \frac{1}{2} \mathbf{j} + 1 \mathbf{k} \right] $.

Key Concepts

Vector Function LimitsTrigonometric LimitsLimit Calculation Steps

Vector Function Limits

When dealing with vector functions, understanding how to compute limits is essential. A vector function is a function that has more than one component, each representing a certain direction. For instance, consider a vector in 3D space defined as \[\textbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j} + h(t) \mathbf{k}\]This vector has three components: $f(t)$, $g(t)$, and $h(t)$. Each of these needs to be handled separately when evaluating limits. To find the limit of the vector function as $t$ approaches a certain value, you must:

Compute the limit of each individual component function. This is similar to finding limits of single-variable functions.
Combine the resulting limits to form a new vector.

By handling each component separately and then combining the results, we preserve the structure of the vector and ensure a correct evaluation of its limit. This approach can be applied to any vector function, no matter the dimension.

Trigonometric Limits

Trigonometric limits are a fundamental aspect of calculus, particularly when dealing with trigonometric functions like sine, cosine, and tangent. These functions often appear as components in vector functions.When calculating trigonometric limits, it’s key to remember the well-known limits:

$\lim_{x \to a} \sin x = \sin a$
$\lim_{x \to a} \cos x = \cos a$
$\lim_{x \to a} \tan x = \tan a$

These limits are straightforward because the trigonometric functions are continuous. This means that you can directly substitute the limiting value into the function. In practice, when dealing with angles converted using fractions (such as $\frac{t}{2}$ in the given exercise), compute the argument first before using these standard trigonometric limits. It makes the process cleaner and reduces room for error.

Limit Calculation Steps

To effectively compute the limit of a vector function, follow these steps meticulously. They ensure that every aspect of the vector is considered, and no components are overlooked.

Step-by-Step Approach

**Identify Each Component:** Break down the vector function into its individual parts. For example, if your vector is given in the form $\textbf{r}(t) = (f(t), g(t), h(t))$, treat each of $f(t)$, $g(t)$, and $h(t)$ as separate entities.
**Compute Individual Limits:** For each component, find the limit as $t$ approaches the given value. Use continuous function properties and trigonometric identities where necessary.
**Combine Results:** Once you have computed the limits for each part, use these to form the vector result of the limit calculation.
**Cross-Check:** Ensure that the results make sense in the given context. Every step from identifying components to combining them should be logical and accurately aligned.

Applying these steps ensures a thorough and accurate limit calculation for vector functions, facilitating better understanding and avoiding mistakes.

Problem 1

Problem 2

Other exercises in this chapter

Problem 1

Find $\mathbf{T}, \mathbf{N},$ and $\kappa$ for the plane curves in Exercises $1-4$ $$ \mathbf{r}(t)=t \mathbf{i}+(\ln \cos t) \mathbf{j}, \quad-\pi / 2

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Problem 1

In Exercises $1-8,$ find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(2 \cos t) \mathbf{i}+

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Problem 2

In Exercises $1-7,$ find the velocity and acceleration vectors in terms of $\mathbf{u}_{r}$ and $\mathbf{u}_{\theta} .$ $$ r=\frac{1}{\theta} \quad \text

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Problem 2

Evaluate the integrals in Exercises $1-10$ $$ \int_{1}^{2}\left[(6-6 t) \mathbf{i}+3 \sqrt{t} \mathbf{j}+\left(\frac{4}{t^{2}}\right) \mathbf{k}\right] d t $$

View solution