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I've attempted many methods of solving this limit problem.

But I feel I'm just guessing now, flailing my arms about like a newborn child.

How would you procedurally solve this problem? I'd like to know where to look first in the future. $$\lim_{t\to 0} \frac{1}{t\sqrt{1+t}} - \frac{1}{t}$$

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  • $\begingroup$ You want to put the expression over a common denominator, and then multiply by the conjugate of the difference involving a square root. $\endgroup$ – user84413 Sep 30 '14 at 19:40
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HINT:

$$\frac{1-\sqrt{t+1}}{t\sqrt{t+1}}=\frac{1-(t+1)}{t\sqrt{t+1}(1+\sqrt{t+1})}$$

Cancel $t$ as $t\ne0$ as $t\to0$

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$$ \lim_{t \to 0} \frac1{t}\left( (1+t)^{-\frac12} -1 \right) =\lim_{t \to 0} \frac1{t}\left( -\frac12 t + \frac38 t^2 +\dots\right) = -\frac12 $$

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