Answer
$\mathrm{i}+3\mathrm{j}-\pi \mathrm{k}$
Work Step by Step
Component by component,
$\displaystyle \lim_{t\rightarrow 1}\frac{t^{2}-t}{t-1}=\lim_{t\rightarrow 1}\frac{t(t-1)}{t-1}=\lim_{t\rightarrow 1}t =1,$
$\displaystyle \lim_{t\rightarrow 1}\sqrt{t+8}=3,$
$\displaystyle \lim_{t\rightarrow 1}\frac{\sin\pi t}{\ln t}=\quad$[$\displaystyle \frac{0}{0}$ ... L'Hospital's Rule].
$=\displaystyle \lim_{t\rightarrow 1}\frac{\pi\cos\pi t}{\frac{1}{t}}=\frac{\pi(-1)}{1}$
$=-\pi$
$\displaystyle \lim_{t\rightarrow 1}(\frac{t^{2}-t}{t-1}\mathrm{i}+\sqrt{t+8}\mathrm{j}+\frac{\sin\pi t}{\ln t}\mathrm{k})= \mathrm{i}+3\mathrm{j}-\pi \mathrm{k}$.
