Answer
$$\lim _{t \rightarrow 1}\left(\sqrt{t} \mathbf{i}+\frac{\ln t}{t^{2}-1} \mathbf{j}+\frac{1}{t-1} \mathbf{k}\right)$$
doesn't exist
Work Step by Step
Given $$\lim _{t \rightarrow 1}\left(\sqrt{t} \mathbf{i}+\frac{\ln t}{t^{2}-1} \mathbf{j}+\frac{1}{t-1} \mathbf{k}\right)$$
Since
\begin{align}
\lim _{t \rightarrow 1} \sqrt t =1\\
\lim _{t \rightarrow 1} \frac{\ln t}{t^2-1} =\lim _{t \rightarrow 1} \frac{\frac{1}{t}}{2t} =\frac{1}{2}\\
\lim _{t \rightarrow 1} \frac{1}{t-1} = \frac{1}{0} =\infty
\end{align}
So, we get
\begin{align}
L&=\lim _{t \rightarrow 1}\left(\sqrt{t} \mathbf{i}+\frac{\ln t}{t^{2}-1} \mathbf{j}+\frac{1}{t-1} \mathbf{k}\right)
\end{align}
doesn't exist
