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Find the limit or show that it does not exist.
$ \displaystyle \lim_{x \to -\infty}\frac{x - 2}{x^2 + 1} $
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02:40
Daniel Jaimes
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 6
Limits at Infinity: Horizontal Asymptotes
Limits
Derivatives
Missouri State University
Campbell University
Oregon State University
University of Nottingham
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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So here we are asked to evaluate specific limit. So we're asked to evaluate a limit as X approaches negative infinity of x minus two over x squared plus one. So with large numbers uh the constant terms are going to become in uh insignificant. So we can ignore the -2 in the plus one. So this would just be the limit as X approaches negative infinity of X divided by x square. So we can eliminate an X. So this would be one over X. For now we're just evaluating the limit as X approaches negative infinity of one over X and we can apply direct substitution here. So this would be negative one over infinity and we know that one over infinity is just to zero in this case. We could have also evaluated this by law rule where we would be taking the derivative of the numerator and the derivative denominator. So the derivative of the numerator would be one, The derivative of the Dye, Romney would be two x. So we would have to apply once again direct substitution in this case. So 1/2 times negative infinity would once again also be zero. So this gives two different ways for evaluating this limit and this is our final answer
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