Show that limt →a 𝐫(t)=𝐛 if and only if for every ε>0 there is a number δ>0 such that if 0<|t-

step 1 So for this proof, we start off with, it's an if and only if statement, so we have to prove both

Show that limt →a 𝐫(t)=𝐛 if and only if for every ε>0 there...

Show that $\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{b}$ if and only if for every $\varepsilon>0$ there is a number $\delta>0$ such that

$$
\text { if } 0<|t-a|<\delta \text { then }|\mathbf{r}(t)-\mathbf{b}|<\varepsilon
$$

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Key Concepts

Vector-Valued Functions

Vector-valued functions take inputs from a set (typically the real numbers) and produce vectors as outputs in a vector space, such as ?^n. The behavior of limits and continuity for these functions is defined by considering their components or using the norm to measure the distance between vectors.

Norm and Distance in Euclidean Space

A norm is a function that assigns a non-negative length to vectors, effectively measuring the 'distance' between vectors in a Euclidean space. When defining the limit of a vector-valued function, the norm is used to ensure that the output vector gets arbitrarily close to the target vector, according to the epsilon criteria.

Epsilon-Delta Definition of a Limit

This concept provides a rigorous way to define the limit of a function. For every positive tolerance, denoted by ?, there exists a corresponding ? such that whenever the input is within ? of a specific point (excluding the point itself), the function's output is within ? of the limit value. It formalizes the idea of getting arbitrarily close to the limit.

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