find-the-limit-of-lim-1-t-1-t-1-t-t-0-

Question Number 90151 by JosephK last updated on 21/Apr/20

f i n d t h e l i m i t o f

l i m \frac{1}{t (\sqrt{1 + t}} - \frac{1}{t}

t \to 0

Commented by abdomathmax last updated on 21/Apr/20

f (t) = \frac{1}{t \sqrt{1 + t}} - \frac{1}{t} \Rightarrow f (t) = \frac{t - t \sqrt{1 + t}}{t^{2} \sqrt{1 + t}}

= \frac{1 - \sqrt{1 + t}}{t \sqrt{1 + t}} w e h a v e \sqrt{1 + t} \sim 1 + \frac{t}{2} + \frac{1}{2} (\frac{1}{2}) (\frac{1}{2} - 1) t^{2}

= 1 + \frac{t}{2} - \frac{t^{2}}{8} \Rightarrow 1 - \sqrt{1 + t} \sim - \frac{t}{2} + \frac{t^{2}}{8}

t \sqrt{1 + t} \sim t (1 + \frac{t}{2} - \frac{t^{2}}{8}) \sim t + \frac{t^{2}}{2} \Rightarrow

f (t) \sim \frac{- \frac{t}{2} + \frac{t^{2}}{8}}{t + \frac{t^{2}}{2}} = \frac{- \frac{1}{2} + \frac{t}{8}}{1 + \frac{t}{2}} \Rightarrow {l i m}_{t \to 0} f (t) = - \frac{1}{2}

Answered by jagoll last updated on 22/Apr/20

\lim_{t \to 0} \frac{1 - \sqrt{1 + t}}{t \sqrt{1 + t}} = \lim_{t \to 0} \frac{1 - (1 + \frac{t}{2})}{t}

\lim_{t \to 0} \frac{- \frac{t}{2}}{t} = - \frac{1}{2}

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