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\noindent {\bf Question}

\noindent Evaluate the limit
\[ \lim_{t\rightarrow\infty} \frac{t^2 +1}{t\ln(t)}.\]


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\noindent {\bf Answer}

\noindent This limit has the indeterminate form
$\frac{\infty}{\infty}$, and so we apply l'Hopital's rule:
\[ \lim_{t\rightarrow\infty} \frac{t^2 +1}{t\ln(t)} =
\lim_{t\rightarrow\infty} \frac{2t}{\ln(t) + 1}. \] The right-hand
limit still has the indeterminate form $\frac{\infty}{\infty}$,
and so we may apply l'Hopital's rule again:
\[ \lim_{t\rightarrow\infty} \frac{2t}{\ln(t) + 1}
=\lim_{t\rightarrow\infty} \frac{2}{\frac{1}{t}}
=\lim_{t\rightarrow\infty} 2t =\infty. \]


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