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Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty}\frac{x^2}{\sqrt{x^4 + 1}} $

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02:30

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Missouri State University

University of Nottingham

Idaho State University

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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Find the limit or show tha…

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Find the limit.$$\lim …

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Find the given limit.$…

So here we have a certain limit where we're evaluating the limit as X approaches infinity of the function X squared over the square root Of X to the 4th plus one. So generally as X approaches very large numbers, constant terms are going to be insignificant. So we can rewrite this as the limit as X approaches infinity of X squared over the square root Of X to the 4th. And since we're taking positive infinity, this would be equivalent to the limit as X approaches infinity of X squared over X squared. And we know that the X squares can cancel out. So this would be equivalent to one as the limit as X approaches infinity and we can also alternately try to solve this by logical rule. However local rule would require many iterations. So for example, Lot little rule asks us to take the derivative of the numerator and denominator. The directive. The numerator would be two x. While the derivative of the denominator would be 1/2 times four x cubed times X to the fourth plus one to the negative three House. So this will require many more generators require at least two restorations to find the limit. And we can just use a simpler method in this case by using our approximation for constant terms and just directly evaluating the limits. And this is our final answer

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