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Find the limit or show that it does not exist.
$ \displaystyle \lim_{x \to \infty} \bigl[\ln (1 + x^2) - \ln (1 + x) \bigr] $
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02:23
Daniel Jaimes
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 6
Limits at Infinity: Horizontal Asymptotes
Limits
Derivatives
Oregon State University
University of Michigan - Ann Arbor
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 evaluate this limit. Not that we can rewrite this into the limit as X approaches infinity of we can combine these using the property of natural log that is Ellen of a minus Ln of B. This is the same as Ellen of a over B. And so using this property we say Ln of one plus X squared minus Ln of one plus X is the same as Ln of one plus X squared over one plus X. And then using limit loss, this is the same as L N of the limit as X approaches infinity of one plus X squared Over one Plus X. And then from here we one up factor of the variable with the highest exponents for the numerator and the same way we will do it in our denominator. And so we get Ln of limits as X approaches infinity of x squared times one plus one over X squared over we have X times you have one plus one over X. And this is equal to L N of we have limits as X approaches infinity of can cancel it. And next here, so we have x times one plus one over X squared over one plus one over X. Now evaluating at infinity we have natural log of we have infinity times one plus one over infinity over one plus one over infinity and we know that One over infinity approaches to zero. And so from here we have Ln of Infinity over one which is the same as Ellen of Infinity and this is just infinity. And so we have shown that the value of the limit is infinity.
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