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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} $

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03:31

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Campbell University

University of Nottingham

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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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.

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