Limits of vector functions Let r(t) = ?f(t), g(t),h(t)?.a.

Chapter 13, Problem 60AE

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QUESTION:

Limits of vector functions Let \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\).

a. Assume that \(\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3}\right)\), which means that \(\lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0\). Prove that

\(\lim _{t \rightarrow a} f(t)=L_{1}, \quad \lim _{t \rightarrow a} g(t)=L_{2}, \quad \text { and } \quad \lim _{t \rightarrow a} h(t)=L_{3} \text {. }\)

b. Assume that \(\lim _{t \rightarrow a} f(t)=L_{1}, \lim _{t \rightarrow a} g(t)=L_{2}, \text { and } \lim _{t \rightarrow a} h(t)=L_{3}\).

Prove that \(\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3}\right\rangle\), which means that \(\lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0\).

Questions & Answers

QUESTION:

Limits of vector functions Let \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\).

a. Assume that \(\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3}\right)\), which means that \(\lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0\). Prove that

\(\lim _{t \rightarrow a} f(t)=L_{1}, \quad \lim _{t \rightarrow a} g(t)=L_{2}, \quad \text { and } \quad \lim _{t \rightarrow a} h(t)=L_{3} \text {. }\)

b. Assume that \(\lim _{t \rightarrow a} f(t)=L_{1}, \lim _{t \rightarrow a} g(t)=L_{2}, \text { and } \lim _{t \rightarrow a} h(t)=L_{3}\).

Prove that \(\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3}\right\rangle\), which means that \(\lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0\).

ANSWER:

Solution 60AE
Step 1 of 2:

a.  Given : which means that  and  
   
r(t) = 〈f(t), g(t),h(t)〉.

If


 

Comparing the components, we get
, and .

Hence it is proved that if  then  , and .

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