Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 29

Determine the $\displaystyle \lim \limits_{t \to 0} \left( \frac{1}{t \sqrt{1 + t}} - \frac{1}{t} \right)$, if it exists.


$
\begin{equation}
\begin{aligned}

& \lim \limits_{t \to 0} \left( \frac{1}{t \sqrt{1 + t}} - \frac{1}{t} \right)
= \lim \limits_{t \to 0} \frac{1 - \sqrt{1 + t}}{t \sqrt{1 + t}}
&& \text{ Get the LCD.}\\
\\
& \lim \limits_{t \to 0} \frac{1 - \sqrt{1 + t}}{t \sqrt{1 + t}} \cdot
\frac{1 + \sqrt{1 + t}}{1 + \sqrt{1 + t}}
= \lim \limits_{t \to 0} \frac{1 - (1 + t)}{t(\sqrt{1 + t})(1 + \sqrt{1 + t})}
&& \text{ Multiply the numerator and the denominator by $1 + \sqrt{1 + t}$ and simplify.}\\
\\
& \lim \limits_{t \to 0} \frac{-1}{(\sqrt{1 + t})(1 + \sqrt{1 + t})}
= \frac{-1}{(\sqrt{1 + 0})(1 + \sqrt{1 + 0})}
= \frac{-1}{(\sqrt{1})(1+\sqrt{1})}
= \frac{-1}{(1)(2)}
= \frac{-1}{2}
&& \text{ Substitute value of $t$ and simplify}\\
\\
& \fbox{$ \lim \limits_{t \to 0} \displaystyle \left( \frac{1}{t\sqrt{1 + t}} - \frac{1}{t} \right) = -\frac{1}{2}$}


\end{aligned}
\end{equation}
$