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Numerade Educator

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Problem 29 Medium Difficulty

Evaluate the limit, if it exists.

$ \displaystyle \lim_{t \to 0}\left( \frac{1}{t \sqrt{1 + t}} - \frac{1}{t} \right)$

Answer

$-1 / 2$

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Video Transcript

to evaluate this limit so that we can rewrite it into the limit. S. T. A purchase zero of one over T. Time squared of one plus t minus The square root of one plus T over tee times the square root of one plus T. Combining the two fractions. We would get limit. SD approaches zero of you have one minus squared of one plus teeth over T times the square root of one plus T. Now in here we would rationalize the numerator and we multiply the whole expression by one plus the square root of one plus T over one plus the square root of one busty which is the conjugate of the numerator. From here we would get limit esti approaches zero of You have one The square of the square of one plus T. Over. We have tee times the square root of one plus T times one plus the square of one plus T. Simplifying this, we would have limit ste approaches zero of one minus one plus T over Teeth and the squares of one plus T times one plus the square root of one plus T simply find further. We would get limit as T approaches zero of negative T over tee times square at the one plus T times one plus the square root of one plus T. And in here we can cancel out the tea and we would get limit S. T. A purchase zero of negative one over The square root of one plus T Times one plus the square root of one plastic evaluating a T. Called zero. We would get negative one over Squared of one plus 0 times one plus the square root of one plus zero, which is just -1/2. And so this is the value of the limits.