Calculus with Vector Functions — Question 1

Question 1 – Solution

Strategy. A vector limit is evaluated component by component. Rewrite each component in terms of a standard scalar limit.

Step 1: First component $$\frac{\sin 3t}{t}=3\frac{\sin 3t}{3t}\longrightarrow 3(1)=3.$$

Step 2: Second component Put $u=2t$. Then $$\frac{e^{2t}-1}{t}=2\frac{e^u-1}{u}\longrightarrow 2.$$

Step 3: Third component Multiply by the conjugate: $$\begin{align*} \frac{1-\cos t}{t^2} &=\frac{(1-\cos t)(1+\cos t)}{t^2(1+\cos t)}\\ &=\frac{\sin^2t}{t^2(1+\cos t)} =\left(\frac{\sin t}{t}\right)^2\frac 1{1+\cos t} \longrightarrow\frac 12. \end{align*}$$ Therefore $$\boxed{\lim_{t\to 0}\mathbf r(t)=\left\langle 3,2,\tfrac 12\right\rangle}.$$

Step 4: Continuous extension Define $\boxed{\mathbf r(0)=\left\langle 3,2,\tfrac 12\right\rangle}$. This makes the function value equal its limit. The vector limit criterion follows because the distance to a candidate vector tends to zero exactly when each coordinate difference tends to zero.