How do you evaluate lim𝐫(t), where 𝐫(t)=⟨f(t), g(t), h(t)⟩?

Name: Problem 5, Chapter 14: Calculus: Early Transcendentals
Uploaded: 2020-06-08T19:19:59-08:00
Duration: 1 min 13 s
Channel: Howard Francis
Description: How do you evaluate lim𝐫(t), where 𝐫(t)=⟨f(t), g(t), h(t)⟩?

step 1 Suppose I'm given the vector valued function, R of T, equals F of T, G of T, H of T, and I want

Question

$$\text { How do you evaluate } \lim \mathbf{r}(t), \text { where } \mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle ?$$

   $$\text { How do you evaluate } \lim \mathbf{r}(t), \text { where } \mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle ?$$

Calculus: Early Transcendentals

William Briggs, Lyle… 3rd Edition

Chapter 14, Problem 5

Instant Answer

Solved by Expert Howard Francis

06/08/2020

Step 1

In other words, if we have a vector valued function $\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle$, the limit as $t$ approaches $a$ of $\mathbf{r}(t)$ is the vector whose components are the limits of $f(t)$, $g(t)$, and $h(t)$ as $t$ approaches $a$. Show more…

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$$\text { How do you evaluate } \lim \mathbf{r}(t), \text { where } \mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle ?$$

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Key Concepts

Component-wise Limit Evaluation

To evaluate the limit of a vector-valued function, you determine the limit of each component separately. This process leverages the property that if the limits of the individual components exist, then the limit of the entire vector function is the vector comprised of these component limits. This method is fundamental when studying continuity and convergence in multivariable calculus.

Vector-Valued Functions

A vector-valued function maps values from a scalar domain (usually time) to vectors in a multidimensional space. Understanding vector-valued functions involves conceptualizing each output as a vector with components corresponding to different dimensions. These functions are essential for describing trajectories, forces, and many physical phenomena in fields such as physics and engineering.

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