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For vector-valued functions of the form $\vec{r}(t)=x(t)\hat{i}+y(t)\hat{j}=(x(t),y(t))$, I learned that the derivative of this function is another vector valued function, $\vec{r'}(t)=\lim_{h\to0}\frac{(\vec{r}(t+h)-\vec{r}(t))}{h}=\lim_{h\to0}\frac{x(t+h)-x(t)}{h}\hat{i}+\lim_{h\to0}\frac{y(t+h)-y(t)}{h}\hat{j}=(x'(t),y'(t))$.

Is the limit of a vector-valued function as the parameter approaches some value defined to be the limit of each of its components as the parameter approaches that value, or is there a definition for these limits that's similar to the epsilon-delta definition of limits that's used for single variable functions?

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1 Answers1

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You can use the Euclidean norm.

Definition of $\lim_{(x,y) \to (x_0, y_0)} (f(x), g(y)) = (a,b)$:

For any $\epsilon > 0$ there exists $\delta$ such that if $(x,y)$ satisfies $\sqrt{(x-x_0)^2 + (y-y_0)^2} < \delta$, then $\sqrt{(f(x)-a)^2 + (g(y)-b)^2} < \epsilon$.

In finite dimensions, the choice of Euclidean norm is arbitrary, since all norms are equivalent.

angryavian
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