Solution Found!
Limits of vector functions Let r(t) = ?f(t), g(t),h(t)?.a.
Chapter 13, Problem 60AE(choose chapter or problem)
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 .