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Numerade Educator

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Problem 20 Easy Difficulty

Find the limit or show that it does not exist.

$ \displaystyle \lim_{t \to \infty}\frac{t - t\sqrt{t}}{2t^{3/2} + 3t - 5} $

Answer

$$
-\frac{1}{2}
$$

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Video Transcript

to find the limit of the following. We first want to rewrite this in powers of T. So this will be the limit as T approaches infinity of t minus. We have t race to 3/2 over. We have to times t raises three halves plus three t minus five. Since we're dealing with limits at infinity, we would factor out the variable with the highest exponents each for the numerator and denominator. So we have the limit. S the approaches infinity of for the numerator, the variable with the highest exponent will be T race with rehab. So we factor out The race to 3/2 and we have T race the negative 1/2 left minus, we have one. And then for the denominator the variable with the highest exponent is T race to three have. So we have to erase the three halves times two plus three times to erase the negative 1/2 -5 times T raised 2 -3/2. And then from here we simplify, we cancel out the common factor and you're left with limit As the approaches infinity uh one over T race to one half minus one over two plus three over t race to 1/2 minus five over T. Races 3/2. And then from here we evaluate A. T equals infinity, And so we have one over infinity -1 over 2-plus 3 over infinity -5 over infinity. Note that Constant over infinity will always approach to zero. And so one over infinity zero three over infinity is zero and five over infinity is always zero, so meaning the limit is equal to -1/2. So this is the value of the limits.