8. Properties of Curves

b. Limits and Derivatives of Vector Functions

Before defining the velocity, we need to define a derivative and before we define a derivative we need to define a limit.

1. Limits of Vector Functions

The limit of a function \(\displaystyle\lim_{x\rightarrow a}f(x)\) is the value the function \(f(x)\) approaches as \(x\) approaches \(a\). For a vector function, we look at the value each component approaches:

The limit of a vector function \(\vec{f}(t)=\langle f_1(t),f_2(t),f_3(t)\rangle\) as \(t\) approaches \(a\), is the vector function whose components are the limits of the components of \(\vec{f}(t)\): \[ \lim_{t\rightarrow a}\vec{f}(t)=\left( \lim_{t\rightarrow a}f_1(t), \lim_{t\rightarrow a}f_2(t), \lim_{t\rightarrow a}f_3(t)\right) \] provided those limits exist.

If \(\vec{f}(t) =\left\langle\dfrac{\sin(t-2)}{t-2},\dfrac{t^2-2t}{t-2},t^2\right\rangle\), compute \(\displaystyle \lim_{t\rightarrow2} \vec{f}(t)\).

\(\displaystyle \lim_{t\rightarrow2} \vec{f}(t) =\left\langle\lim_{t\rightarrow2} \dfrac{\sin(t-2)}{t-2}, \lim_{t\rightarrow2} \dfrac{t^2-2t}{t-2}, \lim_{t\rightarrow2} t^2\right\rangle=\langle1,2,4\rangle\)

Find the limit \(\displaystyle \lim_{s\rightarrow2}\vec{f}(s)\) if \(\vec{f}(s)=\left\langle \dfrac{s^2-4}{s-2},\dfrac{s+1}{3-s},s^2\right\rangle\)

\(\displaystyle \lim_{s\rightarrow2} \vec{f}(s)=\langle 4,3,4\rangle\)

\[ \lim_{s\rightarrow2} \vec{f}(s) =\left\langle\lim_{s\rightarrow2} \dfrac{s^2-4}{s-2}, \lim_{s\rightarrow2} \dfrac{s+1}{3-s}, \lim_{s\rightarrow2} s^2\right\rangle=\langle 4,3,4\rangle \]

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