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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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Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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

Johns Hopkins University