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}$} \end{aligned} \end{equation}$