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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} $

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

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

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this is problem number twenty of the Stuart Calculus eighth edition, Section two point six. Find the limit or show that it does not exist. Limited's T approaches Infinity of the quantity T minus T T minus two times scored a team divided by quantity to temp. See Teo three hours plus three T minus five. Our first step will be to simplify what is inside of the limit. And when we could do that is to first realize this first term tee times t to the square root of team to Times square. To team is tied to the one tempts you to the one half, which is the same a saint to you to the three house. That's our first realization. We take tea to the largest power of all the terms which is tied to the three house, and we're gonna divide each term by teaching three house. So we'LL start with the first term in the numerator. Tea divided by two to the three halves is just one of routine to the one on earth minus two to three years of potatoes. Three houses, just one the denominator. We're gonna t two times t to the tree house a very bright Tito. The three hours which is just too outta St E divided by three to the tea house, which is three over t to the one half my and finally minus five, divided by Tito in three hours. And in this way we are approaching being able to use this definition or this ah result that for tea approach infinity of a function that looks like I want to return to the arm. Where are his rational value? You didn't hear Islam. It is equal zero on. We definitely wanted to use that to our advantage. So we continue on we want he's now properties that limits to separate this limit of a hey function or an expression about him. Another expression on DH. Separate that into a limited then Arabia divided by the limit of the denominator. And then we have a limit of a difference in the mid of sums that we can split up into ice, um, of limits and a difference of limits such there. We'LL be able to separate each of these terms separately into limits themselves. So live limits tapers infinity of one of the t to the one I have Find this Lim as T approaches infinity of one and the denominator will have a limit as t approaches infinity of two plus the limits to purchase infinity of three over t to the one end of minus luminous T approaches Infinity five over TT three halves. And here we use thiss definition over here and we have a few terms that approach. Yeah, zero as T approaches infinity, one of the nominator one in the numerator and too and the denominator and what were left over with zero minus limited as t approaches infinity one which is just one and the denominator way over the limit as t approaches Infinity of two just two zero minus zero. And so our final answer is negative. Wanted to write to or negative one house.

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